
Book_- V 4 ; 



NOTES 



ELE MENTS 



(ANALYTICAL) 

Solid Geometry, 



BY 



CHAS. S. VENABLE. LL.D., 

Professor of Mathematics, University of Virginia. 



NEW YORK: 
UNIVERSITY PUBLISHING COMPANY. 

1879. 



a^l" 



.^3 



\j«; 



In preparing these Notes I have used the treatises of Gregory, 
Hymers, Salmon, Frost and Wostenhohiie, Bourdon, Sonnet et 
Frontera, Joachimsthal-Hesse, and Fort und Schlomilch. 

C. S. V. 



^.J:"^ 339 



In Exchange 
Univ, of V r^. 

Copyright, by 

University Publishing Company, 

1879- 






Notes on Solid Geometry. 



CHAPTER L 



1. We have seen how the position of a point in a plane with ref- 
erence to a given origin O is determined by means of its distances 
^"'M two axes Ox, Oy meeting in O. In space, as there are three 

ensions, we must add a third axis Oz. So that each pair of axes 

rmines a plane, Ox and Oy determining the plane xOy \ Ox 

Oz the plane xOz ; Oy and Oz the plane yOz. And the posi- 

ot the point P with reference to the origin O is determined by 

its distance^ ^M, PN, PR from the zOy, zOx, xOy respectively, these 

distances being measured on lines parallel to the axes Ox, Oy and 

Oz respectively. This system of coordinates in space is called The 

System of Triplahar Coordinates, and the transition to it from the 

System of Rectilinear Plane Coordinates is very easy. We can best 

conceive of these three coordinates of P by conceiving O as the 

corner of a parallelopipedon of which OA, OB, OC are the edges, 

and the point P is the opposite corner, so that OP is one diagonal of 

the parallelopipedon. 

2. If PM = OA ■=L a, PN = OB rr b, PR z= OC = c, the equations 
of the point P are x :=i a, y :=z b, z ^=^ c, and the point given by these 
equations maybe found by the following construction : Measure on/ 
OX the distance OA — a, and through A draw the plane P^AK/A' 
parallel to the plane yOz. Measure on Oy the distance OB = b, / 
and draw the plane pMbR parallel to xOz, and finally lay off 0(y 
= c and draw the plane PMCN parallel to xOy. The intersection 
of these three planes is the point P required. 

3. The three axes Ox, Oy, Oz are called the axes of x, y and z 
respectively ; the three planes xOy, xOz and yOz are called the 



4 NOTES ON SOLID GEOMETRY, 

planes xy, xz and yz respectively. The point whose equations are 
X r= a, X = h, X r= ^ is called the point (a, b, c). 

4. The coordinate planes produced indefinitely form eight solid 
angles about the point O. As in plane coordinates the axes Ox 
and O;^ divide the plane considered into four compartments, so in 
space coordinates the planes xy, xz and yz divide the space con- 
sidered into eight compartments — four above the plane xy, viz. : 
0-xyz, 0-xy'z, O-x'yz, 0-x[yz ; and four below it, viz. : 0'Xyz\ 
O'Xy^z^ O-xy'z', O-x^z', By an easy extension of the rule of 
signs laid down in Plane Coordinate Geometry, we regard all jv's 
on the right of the plane^-s as + and on the left of j's: as — ; all y's 
in front of the plane .V0 as -f and those behind it as — ; all s above 
the plane xy as + and those behind it as — . We can then write the 
points whose distances from the coordinate planes are a, h and c in 
the eiorht different an^Qries thus : 



In the first Octant, 0-xyz P, is 
In the second Octant, Pg is 
In the third Octant, P3 is 
In the fourth Octant, V\ is 
In the fifth Octant, P5 is 
In the sixth Octant, P^; is 
In the seventh Octant, P7 is (- 
In the eighth Octant, Pg is 



The signs thus tell us in which compartment the point falls, 
and the lengths of a, b and c give us its position in these compart- 
ments. 

1. Construct the points i, — 2, 3 ; o, — i, 2 ; o, o, i ; — 4, o, 3 

2. Construct the points i, — 3, — 4 ; 2, — 3, o ; 3, o, — i ; 2, o, o. 

5. The points M, N and R are called the projections of P on the 
three coordinate planes, and when the axes are rectangular they are 
its orthogonal projections. We will treat mainly of orthogonal pro- 
jections. For shortness' sake when we speak simply of projections, 
we are to be understood to mean orthogonal projections, unless we 
state the contrary. 

We will give now some other properties of orthogonal projections 
which will be of use to us. 





{a, 


h. 


c) 


{a, 




K 


c) 


{-a, 


— 


b. 


c) 


(- 


^?, 


h. 


c) 


(''> 


b. 


— 


c) 


(«, - 


K 


— 


c) 


-a, - 


-h, 


— 


c) 


(-a. 


b. 


— 


c) 



NOTES ON SOLID GEOMETRY. 



5 



6. Definitions. 

The projection of a line on a plane is the line containing the 
projections of its points on the plane. 

When one line or several lines connected together enclose a plane 
area, the area enclosed by the projection of the lines is called the 
projection of the first area. 

The idea of projection may be in the case of the straight line thus 
extended: if from the extremities of any limited straight line we draw 
perpendiculars to a second line, the portion of the latter intercepted 
between the feet of the perpendiculars is called the projection of the 
limited line on the second line. 

From this we see that OA, OB and OC (coordinates rectangular) 
are the projections of OP on the three axes, or the rectangular coordi- 
nates of a point are the projections of its distance from the origin on the 
coordinate axes. 

7. Fundamental Theorems. 

I. The length of the projection of a finite right line on any plane is 
equal to the line multiplied by the cosine of the angle which it makes with 
the plane. » 

Let PQ be the given finite straight line, xOy the plane of pro- 
jection ; draw PM, QN perpendicular to it ; then MN is the projec- 
tion of PQ on the plane. Now the angle made by PQ with the plane 
is the angle made by PQ with MN. Through Q draw QR parallel 
to MN meeting QN in R, then QR = MN, and the angle PQR 
= the angle made by PQ with MN. Now MN — QR — PQ cos 
PQR. 

II. The projection on any plane of any bounded plane area is equal to 
that area multiplied by the cosine of the angle between the planes, 

1°. We shall begin with a triangle of which one side BC is parallel 
to the plane of projection. The area of ABC = — BC x AD, and the 

area of the projection A'B'C = - B'C x A'l/ But B'C — BC and /i? 

A'y = AD cos ADM. Moreover ADM = the angle between the 
planes. Hence A'B'C = ABC x cos angle between the planes. 

2°. Next take a triangle ABC of which no one of the sides is pa- 
rallel to the plane of projection. 



6 NOTES ON SOLID GEOMETRY. 

Through the corner C of the triangle draw CD parallel to the 
plane of projection meeting AB in D. Now if we call 6 the angle 
between the planes, then from i"^ A'B D' = ABD cos 6 and B'C'D' 
=iBCD cos e. .-. A'B'D - B'C'D'^: (ABD-BCD) cos d or A^'B'C 
=:ABC cos e, 

3°. Since every polygon may be divided up into a number of 
triangles of each of which the proposition is true — it is true also of 
the polygon, i, e., of the sum of the triangles. 

Also by the theory of limits, curvilinear areas being the limits of 
pol}^gonal areas, the proposition is also true of them. 

8. The projection of a finiie right line upon another right line is 
equal to the first line imiltiplied by the cosine of the afigle between the 
' ^ • ^li7ies\ 

Let PQ be the given line and MN its projection on the line O.v, by 
means of the perpendiculars PM and QN. Through Q draw QR 
parallel to ININ meeting ?M in R. Then PQR is the angle made by 
^ PQ with O.v, and MN =r QR = PQ cos PQR. 

m€tt.^Af^, If there be three points P, P', P" joined by the right lines PP', 
PP" and P'P", the projections of PP" on any line will be equal to 
the sum of the projections of PP' and P'P" on that line. Let D, D', 
D'' be the projections of the points P. P', P" on the line AB. 
Then D' will either lie between D and D" or D" between D and 
D'. In the one case DD" = DD'+ D'D" and in the other DD" — 
DD' — D"D' = in both cases the algebraic sum of DD' and D'D''. 
The projection is + or — according as the cosine of the angle above 
is ^- or — . 

In general if there be any number of points P, P', P", etc., the pro- 
jection of PP'" on any line is equal to the sum of the projections of 
PP', P'P'', etc., or, the projection of any one side of a closed po- 
lygonal line on a straight line is equal to the sum of the projections 
of the other sides on that line. 

lo. Useful Particular Case. 

The projectioji of the radius vector OF of a point P on any line is 
equal the sum of the projections on that line of the coordinates OM, MN, 
PN of the point P. For OPMN is a closed broken line, and 
the projection of the side OP on a straight line must be equal to 
the sum of the projections of the sides OM, MN, and PN on that 
line. 



NOTES ON SOLID GEOMETRY. y 

11. Distance between Two Points. 

Let P and Q, whose rectangular coordinates are {x, y, z) and (.v', 
y\ 0'), be the two points. 

We have from the right parallelopipedon PMNRQ of which PQ 
is the diagonal, PQ' := PM' -f MN^ + QNl But PM ^ x' - x, MN 
=_>/— J/ ; QN = 0'— 0. 

Hence PQ' = {x - xj ^ {y-yy+ (^ - ^J- 

If one of the points P be at the origin then x ^=1 o, y =^ 6, z =^ o, 
and PQ'== .%^'' 4-/^+0''. 

12. To FIND THE Relations between the Cosines of the Angles 

WHICH A Straight Line makes with three Rectangular 
Axes. ■ ^ 

Take the line OP through the origin. Let OP = r, the angle if- ^t<ft^ 
PO.r =: a, POy = /^, PO0 = y, and x\ y, z the coordinates of P. 

Then by Art. 11, r-= .r''+ y''+ z\ 

But, Art. 8, X :=^ r cos a ; y'= r cos ^ ; z ^=^ r cos y. 

Hence r^~ r^ (cos' a -f cos' (5 + cos' y) or .« 

cos' a^ cos' /? + cos' }/ = i. (i) A very im- 
portant relation. 

Cos a, cos /?, cos y determine the direction of the line in rectan- 
gular coordinates, and are hence called the direction- cosines of the line. 
We usually call these cosines /, m and n respectively. So the equa- 
tion (i) is usually written /'+ ^'+ «'= i, (i), and when we wish to 
speak of a line with reference to its direction cosines, we may call it 
the line (/, m, n). Only two of the angles a, (3, y can be assumed at 
pleasure, for the third y will be given by the equation 



cos ;^ = ± Vi — cos' a — cos' //. 

13. We can use these direction cosines also for determining the 
position of any plane area with reference to three rectangular coordi- 
nate planes. For since any two planes make with each other the 
same angle which is made by two lines perpendicular to them respec- 
tively, the angles made by a plane with the rectangular coordinate 
planes are the angles made by a perpendicular to the plane with the 
coordinate axes respectively. Thus if OP be the perpendicular to a 
plane, the angle made by a plane with the plane xy is the angle y ; 
with xz is the angle /i ; and with yz is the angle a. So cos a, cos 
/^, cos y^ are called also the direction cosines of a plane. That is, the 







8 NOTES ON SOLID GEOMETRY. 

direction cosines of a plane with reference to rectangular coordinates are 
the direction cosines of a line perpendicular to this plane, 

14. The relation cos^ a-\- cos^ y3+ cos^ y = 1 enables us to prove 
an important property of the orthogonal projections of plane areas. 
For let A be any plane area, and A^, A,„ A. its projections on the 
coordinate planes yz, zx and xy respectively. Then Art. 7, II., A^ 
— A cos a ; A,, = A cos /? ; A^ = A cos y. 

Squaring and adding we have 

A,2+ A/+ a;- = A- (cos'' a + cos' /S -\- cos' y) 
or a;' + A/ + A/ = A^ 

That is, the square of any plane area is equal to the sum of the squares 
of its projections on three planes at right angles to each other. 

15. To FIND THE Cosine of the Angles between Two Lines in 
Terms of their Direction Cosines (cos a, cos /J, cos ;/) 
AND (cos a\ cos /?', cos }/'). 

Draw OP, OQ through the origin parallel respectively to the given 
lines. They will have the same direction cosines as the given lines, 
and the angle POQ will be the angle between the given lines. 

Let POQ = 6^, OP — r, OQ = /', coordinates of P (a\ )', z), coor- 
*»^^'j6A#/_ dinates of Q (.rV'-s'). 

Wed- or^i, ^^'^ by ^^^' ^ ^' 

•^^^/i/> A,f,'^p PQ"^= {x - xj^ {y -yy + (0 - zj ^ x^ +./+ 0^+ x^^y'^ 

^A^^^*. /^jP' + z'^ ^ (2a;v'+ 2yy -V 2zz), 

>-^ T^f » And from triangle POQ, 

i^.^^S PQ2=: r'^-h r'-- 2/7'' COS 6', 

V;^V^--/ hence r'H r'- 2rr' cos 6 =3 x' -\ r + z' + x'' -^ y" -{- z' - 2 {2xx' 

+ 2y]>' + 2zz'), 
But r'= x' + y''+ z^ and r''= .v"'+,v''-h z'\ 

Therefore rr' cos 6 = au''+ rr'+ zz, 

^ X x' y r z z 
or cos fy = — . — , H — .-^ + — .— . 

r r r r r r 

Hence cos 6 — cos ex cos a + cos fi cos ji + cos ;/ cos ;/' (2) 
which we write cos ^ = //' -f- mm' + nn, (2) 



NOTES ON SOLID GEOMETRY, g 

Cor. 1. If the lines are perpendicular to each other cos /9 = o or 
//'-f- 7)17)1 -{■ 7171 = o (3). (3) is called the condition of perpendicu- 
larity of the two lines (/, w, ti), (/', 711 , 7t'). 

Cor. 2. From expression for cos 6 we can find a convenient one 
for sin^ 6. 

Thus sin- ^ = i — (//'+ /;/;;/'+ 7171)'= (/' + ^/^'^- ?/') (/''+ ?;/"' 

+ 7l'^) — (//'-h 771771 -\- 7l7l'y 

whence sin^ d — {I771 — l'my-\- (Iti — l'7if'\- (77171 — 771 Tif. (4) 

16. To express the distance between two points in terms of their 
oblique coordinates. 

Let P {xyz) and Q [x'y'z) be the two points. 

The parallelopipedon MPQN is oblique. Let the angle XOY 
= A, XOZ = /i, YOZ = y, and the angles made by PQ with the 
axes respectively a, /? and y. Project the broken line PMNQ on 
PQ. This projection is equal to PQ itself. Hence we will have. 

PQ - P:M cos a + MN cos f^ ^- NQ cos y. {a) 

Now project the broken line PININQ on the axes xyz respectively. 
We obtain thus the three equations 

PQ cos a = PM + MN cos A + NQ cos // \ 
PQ cos /3 = PM cos A + MN + NQcos v I (d) 

PQ cos y = PM cos jii + MN cos v + NQ \ 

Now multiply the first of equations (5) by PM, the second by MN 
and the third by NQ and add them taking (a) into account and we 
have 

PQ2 = PM^+ MN^+ NQ^+ 2PM . MN cos A + 2PM . NQ cos 

/A + 2MN . NQ cos V (c) 

or PQ^ =. {x - x'Y + Cr - ^r')^ + (3 - z'y+ 2{x - .v') (y -y) 
cos A +2{x—x')(z — 2')cos j.i-\-2{y—y'){z—z') cos 7^ (5) 

Cor. If one of the points as Q be at the origin then 

PO^ = .V- + r- f s- + 2,\r cos A + 2x2 cos // -f- 2zr cos v. (6) 



17. Di7'Lctio)i Ratios. In oblique coordinates the position of a line 

PM MN NO 
PQ IS determined by the ratios pQ : f3Q-: pTy « and these we 

call direction ratios. \Vc may name these /, w, // respectively, 



lO 



NOTES ON SOLID GEOMETRY, 



taking care to note that we are using oblique coordinates and call 
the line PQ, the line (/, m, n). To find a relation among these 
direction ratios, we divide equation (c) Art. i6, by PQ^ We thus 
have 

I = /^+ m^A- n^ -{- 2/m cos A -f 2/n cos /a + 2mn cos r, (7) the 
desired relation. 

18. The coordinates of the point (xyz) dividing in the ratio m : n 
the distances between the two points {x'y'z) x"y"z" are 



7?ix + nx 

X = ■ , y ■■ 

?n -{- n 



my -\-ny 
m -{- 71 ^ 



. = ^?^'. (8) 



The proof of this is precisely the same as that for the correspond- 
ing theorem in Plane Coordinate Geometr}\ 

19. Polar Coordinates. 

The position of a point in space is also sometimes expressed by 
the following polar coordinates : 

The radius vector OP = r, the angle VOm= 6 which the radius 
vector makes with a fixed axis OZ, and the angle COX which the 
projection OC of the radius vector on a plane yOx perpendicular to 
OZ makes with the fixed line OX in that plane. 

We have OC — r sin ft Hence the formulae for transforming from 
rectangular to these polar coordinates are 

X = r sin cos cp \ 

y z=i r sin d sin qj v (9) 

z = r cos 6 ) 



h 



and these give 



r' = x^ + 1'' + z"- 



tan o) z=.'— 

X 



\ (10). 



Conceive a sphere described from the centre O, with a radius -— a 
and let this represent the earth. Then, if the plane zOx be the 
plane of the first meridian and the axis of z the axis of the earth, 

6 = — ^— latitude, qj ■= longitude of a point on the earth's sur- 



face 



/ 



NOTES ON SOLID GEOMETRY. II 

20. Distance between two points in space in polar coordinates. 

Let Pbe (r', d\ cp) and Q (r, d, cp). Project PQ on the plane xy, 
MN is this projection, draw OM ON the projections of OP and OQ 
respectively on that plane. Through P draw PR parallel to MN, 
then PR == MN. 

And we have 

PQ^ == PR^ + RQ' =: MN^ + (QN - RN)^ 

But in triangle MON 

MN^ rr. OM^ + ON^ - 2OM . ON cos MON, 
or MN^ — r'^ sin^ & + r^ sin^ 6 — 2rr' sin d sin d' cos {(p — cp')* 

Moreover QN = r cos /9 and RN 1= PM = r' cos 6', 

Hence PQ^= r"^ sin^ 6' + r^ sin^ d— irr sin d sin 6' cos (<^ — ^') 

-f (r cos d — r cos ffy 
or 
PQ^= /''^+ r'^— 27'r (cos 6* cos ^' 4-sin d sin ^' cos (^ — <^'))- (^0 



,.^_- 



CHAPTER 11. 

INTERPRETATION OF EQUATIONS. 

TRIPLANAR COORDINATES. 

2 1. Let us take F {x, y, z) — o, that is any single equation con- 
taining three variables .v, y and z. This may be considered as a 
relation which enables us to determine any one of the variables when 
the other two are given. Let these be v and )'. So the equation 
may be written 

z^/{x„y), 

in which we may allege arbitrary and independent values to .v and j'. 
And to every pair of such values there is a determinate point in the 
plane xv ; and if through each of these points we draw a line parallel 
to the axis of 0, and take on it lengths equal to the values of ^ given 
by the equation, it is clear that in this way we will get a series of 
points the locus of which is a surface, and not a solid since we take 
deterviinaie lengths on each of the lines drawn parallel to z. Hence 
F {x, r, z) = o represents a surface in triplanar coordinates. 

22. If the equation contains only two variables as F {x, y)= o then 
it represents a cylindrical surface. 

For F {x, y) = o is satisfied by certain values of x and y inde- 
pendently of 0, and .r and y are no longer arbitrary but one is given 
in terms of the other ; to each pair of values corresponds a point in 
the plane xy, and the locus of these points is a curve in that plane. 
If through each point in this curve we draw a coordinate parallel to 
z, every point in that coordinate has the same coordinates .v and >• as 
the point in which it meets the plane xy. Hence F {x,y) — o repre- 
sents a surface which is the locus .of straight lines drawn through 
points of the curve ^{x,y) = o in the plane xy and parallel to the 



NOTES ON SOLID GEOMETRY. 1 3 

axis of 0. This locus is either what is called a cylindrical surface 
with axis parallel to sr or a plane parallel to the axis of z according 
as the equation F {x^y) = o in the plane xy represents a curve or a 
straight line. 

For example, x'^ -{■ y^ — r^= o in rectangular coordinates is a 
right cylinder with circular base in plane xy (since x^-^y^— r^ is a 
circle in plane xy) and its axis parallel, to the axis of z. 

And dx •\- by — c ^=: o \^ 2i plane parallel to the axis of 0, intersect- 
ing the plane xy in the line ax -\' by =^ c. 

Similarly F {x, z) — o represents either a cylindrical surface with 
axis parallel ioy or a plane parallel ioy, ^ 

F (y, 0) = o represents either a cylindrical surface with axis parallel 
to the axis of :r or a plane parallel to this axis. 

23. An equation containing a single variable represents a plane or 
planes parallel to one of the coordinate planes. 

Thus X =^ a represents a plane parallel to the plane j/0. 

And 2i?>/{x) = o when solved will give a determinate number of 
values of :r, as x ^= a, x =. b, x = c, etc., so it represents several 
planes parallel to the coordinate planej^^. 

Thus also F {y) = o represents a number of planes parallel to the 
plane xz. 

And F {z) = o, a number of planes parallel io xy, 

24. Thus we see that in all cases when a single equation is inter- 
preted it represents a surface of some kind or other. 

The apparent exceptions to this are those single equations which 
from their nature can only be satisfied when several equations which 
must exist simultaneously are satisfied. As for example 

(.r — ay + (jF — by -\- {z — cY — o. This equation can only be 
satisfied when [x — a)^ = o, {y — by = o, {z — cY = c, or x = a, 
y ~ b, z = c. 

Now these represent three planes, but being simultaneous they 
represent the point a, b, c. 

So also {x — aY -\- {y — by '= o is only satisfied hy x z=z a, y =z b, 
and hence though x = a Is a plane, and y = b is 3. plane, the two 
together must represent a line common to both of these planes, that 
is their line of intersection, which must be parallel to z, 

25. In general two simultaneous equations as 

/{x,y, z) = F {x,y, z) = o 



14 NOTES ON SOLID GEOMETRY. 

represent a curve or curves, the intersections of the two surfaces 
represented by the two equations. 

Thus ___.[■ taken simultaneously we have seen represent a straight 

line parallel to the axis of z, the intersection of these two planes. 

F (x) = o ) 

^ ^ _ >■ represent a number of straight lines parallel to the 

axis of z, the intersections of the several planes parallel respectively to 
the planes j'0 and xz. 

■r^\ ! _ [ represent a number of straight lines parallel to the axis 

of j^, etc. 

F ixy) =. o ) 

T^ / ; >• represent the curves of intersection of the two cylin- 

F {xz) = o ) ^ -^ 

ders F [x, y) = o and F {y, z) — o, etc., etc. 

26. Three simultaneous equations 

F {x, y, z) =0 \ F {x, y) = o\ 

as /{x, y^ z) =:ol or F (x, z) = o i etc., 

(p{x,y,z) = 0) F(j^, 0) --=0 ) 

represent points in space or the intersections of the lines of intersec- 
tion of the surfaces. 
The simplest case is, 

X =^ a \ 

y ^^ ^ \ representing the point (^, b, c),. 
z := c ) 
So also 

x^ + y^ = 2z^ \ 

•X -{- y =^ 2z V represent points which can be found by 

xy = 4- ) 

solving the three equations which themselves represent different sur- 
faces. 

Interpretation of Polar Equations. 

27. 1°. r ■= a represents a sphere having the pole for its centre. 
Hence the equation F (r) = o which gives values for r as r :=: a, 
r ^= b, r =: c, etc., represents a series of concentric spheres about the 
pole as centre. 



NOTES ON SOLID GEOMETRY. 



15 



2°. 6 =^ a represents a cone of revolution about the axis of 2: with 
its vertex at the origin of which the vertical angle is equal to 2a, 
Hence the equation F (^) = o giving values 6 — a, 6 = ^, etc., 
represents a series of cones about the axis of 3:. having the origin for 
a common vertex. 

3°. (p =1 ^ represents a plane containing the axis of ^ whose line 
of intersection with the plane xy makes an angle a with the axis of 
X. Hence the equation F (^) = o which gives values cp=. ^, qj 
= /?', etc., represents several planes containing the axis of inclined 
to the plane zOx at angles /?, /5', etc. 

4°. If the equation involve only r and ^ as F (r, 6) = o, since 
F {r, 6) = o gives the same relation between r and 6 for any value 
of q), it gives the same curve in any one of the planes determined by 
assigning values to cp. Hence it represents a surface of revolution 
traced by this curve revolving about the axis of z. 

Example, r = a cos 6 is the equation of a circle in the plane 
xz, or in any plane containing the axis of z. - Hence r z= a cos 6 
represents a sphere described by revolving this circle about the axis 
of -sr. 

5°. If .the equation be F((^, 6) = o for every value of q) there 
are one or more values of 6 corresponding to which lines through 
the pole may be drawn, and as cp changes or the plane fixed by it 
containing O0 revolves, these lines take new positions in each new 
position of the plane, and thus generate conical surfaces about Oz. 
(A conical surface being any surface generated by a straight line 
moving in any manner about a fixed straight line which it inter- 
sects. ) 

6°. If the equation be F (r, q?) = o, for every value of q? there are 
one or more values of r, thus giving several concentric circles about 
the pole in the plane determined by the assigned value of q). As q? 
changes, or the plane through Oz revolves these values of r change, 
and the concentric circles vary in magnitude. The equation thus 
represents a surface generated by circles having their centres at the 
pole, which vary in magnitude as their planes revolve about the axis 
of 2: which they all contain. 

7°. If the equation be F (r, 6, qj) = o, it represents a surface in 
general. For if we assign a value to q) 3.s qj = /3, then F (r, 6, /3) 
=: o will represent' a curve in the plane q? =: j3. And as <p changes 
or the plane revolves about Oz this curve changes, and the equation 
will represent the surface containing all these curves. 



1 6 NOTES ON SOLID GEOMETRY. ^ 

28. Two simultaneous equations in polar coordinates represent a 
line, or lines — the intersections of two surfaces. And three simulta- 
neous equations represent a point or points— the intersections of three 
surfaces. 

Thus 

r zn a \ 

6 = a\ taken simultaneously represent points determined 
by the intersection of a sphere, cone and plane. 



CHAPTER III. 

EQUATION OF A PLANE. 

COORDINATES OBLIQUE OR RECTANGULAR. 

29. To find equation of a plane in terms of the perpendicular from the 
origin and its direction cosines. 

Let OD = /) be the perpendicular from the origin on the plane, 
and let it make with the axes Ox, Oy and Oz the angles a, /? and y 
respectively. Let OP be the radius vector of any point P of the 
plane ; OM, MN and NP the coordinates of P. 

The projection of OM + NM + NP on OD is equal to the pro- 
jection of OP on OD. 

The projection of OP on OD is OD itself, and the projection of 
OM + MN + NP on OD is x cos a -^-y cos p + z cos y. 

Hence we have x cos a -i-y cos /3 -\- z cos y •= p. (12) 

30. To find the equation of a plane in terms of its intercepts on the 
coordinate axes (coordinates oblique or rectangular). 

Let the intercepts be OA = a, OB — b, OC = c. The equation 
(12) may be written 

X y 2 



p sec a p sec ^ ps^c y 

But since ODA, ODB and ODC are right-angled triangles, we have 
/> sec or = OA = a,^p sec /S = OB — h, p sec y = 0C — c. 
Therefore the equation becomes 

X y z , . 

-+! + -=: (13) 

the equation of the plane in terms of its intercepts. 



1 8 NOTES ON SOLID GEOMETRY. 

31. Any equation Ax + By + Cz = D (14) of the first degree in x, 
y and z is the equation of a plane. 

For we may write (14) 

X y z _ 

A B "C" 

And putting -^=: a, -^ = <^, — = ^, we have the form (13). 

Hence (14) is the equation of a plane in oblique or rectangular 
coordinates. 

Hence to find the intercepts of a plane given by its equation on 
the coordinate axes, we either put it in the form (13) or simply 
make^ =r o and =: o to find intercept on x \ 2 = and jr = o to 
find intercept ony ; x= o and j' =: o to find intercept on z. 

Example. Find the intercepts of the plane 2.r + j)/ — 52 = 60. 

32. It is useful often to reduce the equation Ax -\- By -{- Cz = D 
to the form x cos a -{-y cos /3 -{- z cos y =p in rectangular coordi- 
nates. We derive a rule for this. 

Since both of these equations are to represent the same plane, we 
have 



cos a _ cos /? _ cos y __P __ Vcos* a + cos^ f6 + cos^ y 
A ~ B " C ""D"^ A^ + B^ + O 



I 



Va' + b^+ o 



1:T a ^, B 

Hence cos a -=. — , cos p : 



C D 



Va" + 6^ + 0" Vam-bm-o 

Hence if we write (14) 



A B 

X + , y + 



VA'+B' + C^' -v/A' + B^ + C^ VA' + B^+a 

D 



VA''+ B^ + Q^ 
it is in the perpendicular form (12). 



(15) 



NOTES ON SOLID GEOMETRY. 



19 



Hence the Rule: If ive divide each term of the equation Ax + By 
+ Cz=:D, hy the square root of the sum of the squares of the coefficients 
of X, Y and z, the new coefficients will be the direction cosines of the per- 
pendicular to the plane from the origin^ and the absolute term will be the 
length of this perpendicular. 

Example. Find the direction cosines of the plane 2x + ^y — 4z 
= 6 and the length of the perpendicular from the origin. 

Result. 

2 23 —4 

cos a = — = — -:=, cos p = — -=, cos y — 



^4 + 9 + 16 V 29 V 29 V 29 



V 29 

33. To find the angle between two planes (coordinates rectangu- 
lar). 

If the planes are in the form 

X cos a -^ y cos /S ■\- z cos y ^=^ p 

X cos a^ -\- y cos /5' + cos y = p\ 

then since this angle is equal to the angle of two perpendiculars from 
origin on the planes the cosine will be (Art. 15) cos V= cos a cos a' 
+ cos ^ cos fS + cos y cos y' , 
If they are in the form 

A^ + B^ + C0 = D 
A'jr + B> + Cz ^ D\ 

A B 

Then cos a — — , cos (5 



^A^+ W A-O VA2+ B^ + C^ 

C 



cos y- 



VA^ + B^ + C^ 



cos a = — - , cos p 






cos y 



y- A 'i + B'^ + C'^ 



A A .r AA' + BB' + CC , ., 

And rns V = — (i6) 

VA= + B' + CVA" + B"^ + C"' 



20 NOTES ON SOLID GEOMETRY 

From this 
. , ,, (A'+ B'+ C°) (A^^+ B"+ Cn -(AA'+ BB'+ CC')' 
^^■^ (A''+ B^ + a)(A'^+B'^+C'») 

. , (AB'- A'B)^+ (AC-- A-C)'+ (BC^- B'C)' 

. sin V _ (AH B^ + e)(A'='+ B'^+C'^) " ^'' 

Cor. I. If the planes are perpendicular to each other, then cos y=o. 
.•. AA'+ BB'+ CC = o {\%^ is the condition of perpendicularity of the 
planes. 

Cor. 2. If the planes are parallel sin V = o. Hence 

(AB' ■^- A'B)5 = (AC - A'C)^ = (BC - B'C)= = o 
or AB' - A'B = o AC - A'C = o BC - B'C = o ' 

ABC., 

or the condition that the two planes shall be parallel, is that the coefficients 
o/x, Y and z in the two equations shall be proportional. 
Ex, I. Find the angle between the planes 



B-S^^ ^ , ^^^^ 



ffJ^T^j" 7^""^^ ^+ ^I' + S^^ 5 and 3a;-4y-} 2= lo. 

2. Show that the planes 

^ ^^~ :t: + 3jV — 5^ = 20 and 2x -{• y •{• z ^=^ lo are perpen- 

dicular to each other. 

3. Write the equation representing planes parallel to the plane ^x 

34. To find the expression /or the distance from a point P (x'y'z') to a 
plane (coordinates rectangular). 

1°. Let the equation of the plane be of the form 

:v cos o' + j^^ cos /3 + ^ cos ;K = / when p = OD. 

Pass a plane through P parallel to the given plane, and produce 
OD to meet it in D'. 

The equation of this plane will be 

x' cos a -i-y cos ft -\- z' cos y — p' when OD' =/. 

Now let PM be the perpendicular from P on the given plane. 
Then PM = OD' - OD ==/ -/. 



NOTES ON SOLID GEOMETRY, 21 

Hence PM •=. x' cos a -{• y' cos ^ -\- z' cos y — p. 

And X {x' cos a -{-/ cos /3 + z' cos y) —p (20) is the expression 
for the perpendicular from the point x'y'z' on the plane x cos a^ y 
cos y5 + cos y — p, the sign being -f or -— according as / is on the 
side of the plane remote from the origin or next to it. 

2°. Let the plane be in the form Kx -f By -f- C0 = D. 

Then cos a — — = etc., etc. (15) Art. ^2. 

^A^ + B^-hC^ 

Hence the expression 

±{x' cos a i-y' cos ^ -^ z' cos y ^ p) becomes 



Ex. Find the length of perpendicular from the point (3, 2, i) on 
the plane 

3.r -[- 4.y — 6z = 24. 

T^i 9 + 8 — 6 — 24 —13 

Result. p — ^ ^ — _j_ . 

V9+ 16+ 36 V 61 

*35. The equation of the plane in the form .rcos a -\-y cos j3 -{- z 
cos y = p may be used to demonstrate the following theorem in 
projec'ions. 

The volume of the tetrahedron which has the origin for its vertex and 
the triangle ABC/^r its base is equal to the three pyramids which have any 
point (x, y, z) in the plane ABC for their common vertex and for bases 
the projections of the area ABC on the three rectangular coordinate planes 
respectively. 

For let A be the area of the triangle ABC and 

X cos a ■\- y cos ^ -{• z cos y ^= p 
the equation of its plane. 

Multiply this equation by A. 

Then A cos a . x -\- \. cos (3 .y + A cos y . z := Kp 

or -^A cos a , X -\- \K cos (3 .y + ^A cos y , z =i ^A/. 

But A cos a, A cos /?, A cos y, are the projections of A on the 
planes j'2r, xz, and xy respectively, and Xy y and z are the altitudes of 
the tetrahedrons which have these projections as bases and the point 
{x, y, z) as common vertex, and -|-A/> is the volume V of the pyramid 



22 NOTES ON SOLID GEOMETRY, 

which has the origin for vertex and A for base. Hence the theorem 
is true. 

Calling these projections A^, A^, and A^, we may write the equa- 
tion of the plane k^x + KyV + A^ = 3 V. (22) 

36. To find the polar equation of a plane. 

Let OP = r, POS = 6, YQW = cpbe the polar coordinates of a 
point P of the plane. 

Let OD = a = perpendicular on plane ; angle DOS = a, D'OM' 
= /3, and POD = go. 

Then -^-p- = cos POD = cos Ce9, or — = cos go. Now in order to 

^ express oo in polar coordinates conceive a sphere about O as centre 

with OP = r as radius. Prolong OD to D^' on the sphere. Draw 
-^^'^'^'^-^the arcs of great circles SPF, SD"D; MP'D' and D'T. 
•c^c^M.^.,^s^^ The triangle SD'T has for its sides SD" =: o', SP = 6, D"P — go 
Smt "^^l^nd angle D"SP = D'OF = ft -^ cp. But 

T£f% r^^^ ^Og J^n^ ^ ^^g gj)„ ^^g gp ^ g.^ gj^,r gjj^ gp ^^g j),,gp^ 



i«*J2.. 



Or 



"^^ cos Gj? = cos a cos ^ + sin ar sin 6 cos {/S — cp). 

Therefore — = cos a cos ^ + sin ^ sin 6 cos (/J — ^) (23) is the 
polar equation of the plane. 

37. The general equation of the plane Ax + B>' + C0 = D may 
be reduced to the form 

A'x + BV + Cz= I (24) by dividing by the absolute term D. 

And also to the form 

z m mx 4- ^+ ^ {25) by dividing by C — transposing and putting 

— — z=z m, —— = n and -— = c. These two forms are very useful in 

the solution of problems and in finding the equations of the plane 
under given conditions. 

Plane under Given Conditions, 

38. 1°. The equation of a plane through the origin will be of the 
general form Ax + By + C2: = o, for the equation must be satisfied 
by :r = o, j^ = o and = 0. 

2°. The equation of a plane which contains the axis of z is of the 



r 



NOTES ON SOLID GEOMETRY. 



23 



form Kx + By = o ; a plane containing the axis oi y is Kx + Cz 
= o ; one containing the axis of x is By -}- Cz =^ o, 

3°. The equation of a plane parallel to the axis of z is Ax + By 
= D ; of one parallel to the axis of j^ is Ax -\- Cz = T) ; one parallel 
to thQ axis of xisBy + Cz = J). 

.4°. The equation of a plane parallel to the plane j^0 is Ax = D ; 
parallel to ^2: is By = D ; parallel to xy is Cz == D. 

These equations we have had already in the forms x= ±a, y=z ±^, 
z = ± c» 

39. To find the equation of a plane containing a given point (a, b, c) 
and parallel to a given plane Ax + By + Cz = D. ( i ) 

First, since the required plane is to be parallel to (i) it may be writ- 
ten Ax •\-By -\-Cz-=^V>' (2) when D' is undetermined. Secondly, the 
coordinates (a, <5, r) must satisfy (2). Therefore A^-hB^ + Q =D\ 
Hence by subtraction we eliminate D' and obtain 

A(^ — a) + B (jF — <^) + C (0 — ^) = o or 

Ax + By -\-Cz— Aa-\-Bh ^-Cc (26) 

the required equation. 

Example. Find the equation of the plane passing through the 
point (i, 2, 4) parallel to the plane 2x ■\- ^y — ^z ^=:. 6. 

40. To find the equation of a plane passing through three given points 
(x', y', z'), (x", y", z") and (x'", y'". z'"). 

Let the equation of the plane be of the form Ax -f By + C0 = i, 
A, B and C to be determined by the given conditions. 

Since the plane is to contain each of the points, we must have 
A^ + By' + C^^ =1 

A^'"+By'^+a'"=i. 

Hence 



^- z 



A = 






x,y, z 

It tt II 
X ,y , z 



B= 



X, I, z' 
x'\ I, z" 
x'\ I, z" 



X ,y, z 

n It II 

X ,y , z 

III III I. 



C= 





x',y, I 






x",y", I 






x", v'", I 




1 ' 1 

x,y, z 




x\y\ z" 




J 


<:'",/", z'" 





X ,y , z 
Substituting these values in the equation Ax + 'Ry -\- Cz =■ i , 



24 



NOTES ON SOLID GEOMETRY. 



we have 






X + 



I, s 

I, z' 



I, 



J + 



^,J^^, I 



y 



X 



ttt ,n f 

X ,y , 



z = 



x,y, z 
x"\ y"' z 



. (27) 



Mjk4. 



But from plane coordinate geometry the coefBcients oi x^y ahd z 

(jt^^Z"*)^"^ these equations are the double areas of triangles in the planes j^^, 

jrji^\/^^z and xy respectively. Moreover these triangles are the projections 

j ^^ ^^ triangle of the three given points, on these planes. Hence 

-2r£x — ^ comparing this equation wi h the equation (22) 



we see that 



K,x + Ky 4- Kz = 3V 



JiLy 



II II 



= 6V, That is = 6 times the volume of the 



pyramid which has the origin for vertex and the triangle of the three 
given points for base. This equation fully written out is 

x {y'z"'-y"z")+x"{y"z'-yz"') +x"'iyz"-y'z')=^6v. (28) 

41. To fi7id the equation of the planes which contain the line of inter sec- 
tion of the two planes Ax + By + Cz = D and Ax' + By'+ Cz' = D'. 

ThisequationisAjr + By-f C0 — D + K( A'^ + B> + C'0 — D')rr:o(29) 

when K is arbitrary. For this represents a plane when K takes a 
particular value and it is sa'isfied when A^ + Bj' + C^ — D = o and 
A'x + ^y + C'z — D' =: o are satisfied simultaneously. Hence it 
is a plane containing their line of intersection. Hence as K is arbi- 
trary it (24) represents the planes containing the line of intersection 
of the two given planes. 

42. When the identity KU + KiUi + KgUg = o (30) exists between 
the equations U = o, U2=o, Us^ o of three planes, then these planes 
intersect each other in one and the same straight line. This is an 
easy corollary of Article 41. Also when the equation of[ the first 
degree in x, y and z contains a single arbitrary constant all the 
planes which it expresses by assigning particular values to this con- 
stant intersect each other in one and the same straight line. This 
line of intersection may be at infinity and then the planes are all 
parallel. 

Example i. The planes represented by the equation ()x^My-\-2z 
z= 3 (fS^ arbitrary) all contain the line of intersection of the two 
planes 6a: + 22^ — 3 — o 2indy= o. , 



NOTES ON SOLID GEOMETRY. 25 

Example 2. The planes represented by 2x -\- ^y — 42 = n {n 
arbitrary) are parallel. 

Example. The planes 2>^ -\- \y ■\- 6z — 2 \ 

4X + 4y -{- 6z = 2 J 
intersect in one and the same straight line because 

{sx-^4y + 62—2) — {x + 2y+s2 — i) -^ ^{4^ + 4y + 6z—2) = o 

is an identity. 

43. Wken between the equations of four planes in any form U = o, U] 
= o, Ug — o, U3 = o the identity 

KU + K1U1 + K2U2 + K3U3 = o (31) exists, then these four planes 
intersect each other in one and the same point. For then any coor- 
dinates which satisfy the first three U = o, Uj = o and U2 = o will 
satisfy the fourth U3 = o. 

44. Example i. 'Find the equation of the plane passing through 
the origin and containing the line of intersection of the two planes 
Aa^ + By + C0 = I and h!x + B> + Cz — i. 

First we have Kx + By + C0 — i + K (A'jtr + B> -^ Cz —\)—o 
for all the planes containing the line of intersection of the two given 
planes. But as the required plane must contain the origin, the 
equation must be satisfied by (o, o, o). Hence we have — i— K=o. 
.-. K = - I. 

The required equation is therefore 

A.T + By + C^ - I - {Mx + B> + Cz - i) = o 
or (A - A') :r -f- (B - B')^ + (C - C) := o. 

Ex. 2. On the three axes of x, y and z take OA = a, OB =: b, 
OC = c and construct on these a parallelopipedon having MP as the 
edge opposite parallel to OC, and AR in the plane xz the edge 
opposite and parallel to BN. 

Find the equation to the plane containing the three points M, N 
and R. 

X V 

Now NR is the line of intersection of the two planes — +— - = i and 

a b 

z 

— = 1. Hence the plane containing this line must be of the form 



-/ 



26 KOTES ON SOLID GEOMETRY. 

'X' V / z \ 

— h-T— i+K{ ij=:o. To determine K we impose the 

condition that this plane shall pass through the point jNI {a, b^ o). 

•* - / Hence we have — + - — i -f- K ( ^i ) == o. .-. K=: i. 

^ a \c J 

Therefore the required plane is 

XV z X y z 

-- + V — H 1 — o or h^H — = 2. 

a c a c 



9C) |,| 



Ex. 3. Find in like manner the equation of the plane containing 
the points P, B, and C, in the same figure. 



V Z X 

Result, -y-+ : 

o ^ ^^/y^/-/ oca 



•• ^^ ^^^ 45. If two given planes be in the normal form as 

X cos a:4-j^cos/5 + cos ;/=/> and.r cos a! -\-y cos fi' •\-zzo^ y-=zp\ 

The plane containing their line of intersection is 

X cos a-\-y cos fi-\-z cos y — p-\-Y^{x cos a ■\-y cos ^' + Z cos y' 

-p') = o 
And if K = ± i the equation becomes 

. X cos a -\-y cos /? + cos y — P±l{x cos a + y cos /?' + cos y' 

-/)=-• o 

\which represents the two plane bisectors of the supplementary angles 
imade by the given planes. 

That is to find the equations to the plane bisectors of the supplementary 
> angles made by two given planes, put their equations in the normal form 
* . and then add and subtract them. 

Example. Find the two planes which bisect the supplementaiy 
: angles made by the planes 2x-^'^y'\-z = 5 and ^x-{-4y—2z = 4. 

Result, 2^ + 3^+5-5 j^ 3^ + 4>'-2.-4 ^^, 

VI4 V2I 

Remark, If we place A = jv cos a-\- y cos ji -\- z cos y — p and 
.A' = X cos a' -\- y cos ^' -{- z cos y' — /'. 

Then A'— A = o is the plane bisector of one of the angles be- 
itween the planes A and A' and A + A' = o is that of the supple- 
imentary angle. 

46. The three planes which bisect the diedral angles of a triedral have 
a common line , of intersection. Let A = o, A'= o and A" = o be three 



NOTES ON SOLID GEOMETRY, 



27 



planes in the normal form, and let the origin be within the triedral angle 
formed by the three of which P their point of intersection is the vertex. 

Then the plane bisectors of the angles made by these planes is 
A — A' = o, A"— A =3 o, A'— A" = o. And as these when added 
together vanish simultaneously, it follows that these three planes 
have a common line of intersection. 

We can give this theorem another form by conceiving a sphere to be 
described about the vertex of the triangular pyramid as a vertex. The 
three planes A= o, A':= o, A"= o cut the surface of the sphere in 
arcs of great circles which form a spherical triangle and the three 
planes A — A'=o, A''— A = o and A'— A'' = o cut the sphere in 
three arcs of great circles which bisect the angles of this spherical 
triangle and their common line of intersection pierces the sphere in 
the common intersection of these arcs. Hence the above demon- 
strates the following theorem, namely. The arcs of great circles which 
bisect ihe angles of a spherical triangle cut each other in the same point 
(the pole of the inscribed circle of the triangle). 

^J. To find the point 0/ intersection of the planes Kx + By + C^r 
^ D, Kx + B> + Cz = D', A'x' + B'> + C'2 = D". 

We have by elimination 



X = 



D, 


B, 


C 


D', 


B', 


C 


D" 


B", 


C" 


A, 


B, 


c 


A', 


B', 


c 


A" 


B", 


C" 



y 



A, 


D, 


C 


A', 


D', 


C 


A" 


D" 


, C" 


A, 


B, 


c 


A', 


B', 


c 


A" 


B", 


C" 



A, 


B, 


D 


A', 


B', 


D 


A", 


B", 


D" 


A, 


B, 


C 


A', 


B', 


C 


A", 


B", 


C" 



(32) 



Hence the condition that one of these shall be parallel to the line 
of intersection of the other two, or that the planes shall not meet in 



a point, is ac, ^ X - 


A, B, C 




A.t.C^4''UA'.BUB".'f'*A 




A', B', C 
A", B", C" 


= 0, that is '^^ 


A(B'C"-B"C') + A'(B"C - BC") + A'(BC'- B'C) = o. 


47. The condition that four planes 


A;«r + I 
A'jt: + 
A";«r + 
A"';^^^ 


B^ + Cs + D 
B> + C's + E 
B'> + C"2 + 
-B"> + C"'2 


= 
>' =0 
-D" =0 
+ D"' = o 


shall meet in a point is' 



28 



NOTES ON SOLID GEOMETRY, 



a; b, c, d 

A', B', C\ D' 
A", B", C\ D" 
A"', B'", C", D'" 



= o. (33) 



49, We have seen that the equations of two planes Kx-\-^y ^Cz 
— • D = o and h!x-{- Bj^ + QJz — D'z= o added together one or both 
of them multiplied by any number give the equation of a plane 
which contains the line of intersection of the two given planes. If 
we combine these two equations so as to eliminate x we shall obtain 
a plane parallel to the axis of x, containing this line of intersection. 
If we eliminate _)^ we obtain a plane parallel to the axis of j contain- 
ing the same line ; and finally if we eliminate z we obtain a plane 
parallel to the axis of containing the" same line. 



CHAPTER IV. 

THE STRAIGHT LINE. 

50. T/ie equations of any two planes taken simultaneously represent 
their line 0/ intersection, 

;;, ^, ^, t (34) represent a straight line the co- 

ordinates of every point of which will satisfy the two equations. 

If we eliminate alternately x and y between these equations we 
obtain equations of the form 

~ I (35) ^wo planes perpendicular respectively to the 

planes xy and yz which represent the same straight line as equations 
(35). These non-symmetrical forms (35) are very useful. The 
planes x = mzi-a, y = nz + d are called the projecting planes of the 
line on the planes of xz 2ind yz, and these equations are also the: 
equations of the projections of the line on those planes respectively,. 

y /J A^ t) n n 

If we eliminate we get - — -=i ^or yz=z — x-]-q p the equa-- 

71 in -^ VI 771 

tion of the projection of the line on the coordinate plane xy. 

The equations (35) of the straight line contain four arbitrary con- 
stants, VI, n, p, q^ to which we can give proper significance by com-- 
paring these equations with the equation;^ = 77ix-\-h in plane coor- 
dinate geometry. 

The equations (35) maybe thrown in the form 

^^=.2JZf=f (36) 

7n n I ^^ ^ 

which gives us an easy choice of fixing the line by the equations; 
of any two of its projecting planes. 

51. To find the equations of a straight line in te7'vis of its directiom 
cosines a7id the coordinates a, b, c of a point 07i the li7ie: 

29 



30 



NOTES ON SOLID GEOMETRY. 



Let a, (3, y be the angles made by the line with the coordinate axes 
respectively. Let / be the portion of the line between any point 
{x, y, z) on the line and the point (^, h^ c). Then / cos a = x—a ; 
I cos /? ^=.y—b ; / cos y— z—c ; and eliminating / we have 

x—a y—b z — c , . 

COS ^ COS/i COS)/' ^^'^ 

This form (37) of the equation of a straight line is symmetrical 
and is therefore very useful. It contains six constants but in reality 
only four independent constants, since the relation cos^ ^H-cos^ y3 
+ cos^ y :=• \ holds, and of the three a, b, c one of them may be 
assumed at will, leaving only two independent. 

We have seen that the equation (35) may be thrown into the form 
KZl^' So also (37) may be thrown into the form (35) by finding 
from them expressions iox y and x in terms oi z. 

52. To find the direction cosines ofajiy straight lijie given by its eqicatiojis. 

If the equations be in the form 

X — a y — b z — c 

— ^^ — = —^ — = — rr — . L, M and N are proportional to the 
L J\I ]N ' ^ ^ 

direction cosines of the line. 
So that we have 



cos a _ cos /? _ cos y _ Vcos' a + cos^ p + cos^ y 



L 


M 




N 




VL^ + MHN^ 




~V^ 


+ M^ + N2 


Hence 




















L 




cosfJ= 


M 


cosy 




N 




VL~- 


fxM^ 


+ N=^ 


^L^ + M^+^N^' 


~VL 


•^4-M^-fN^ 



(38) 

Hence to find the direction cosines of any straight line 

Ax -\-By + Czz=zJ) ) 

K'x + B> + C^^ = D' j 

we throw the equations into the form 

x — a y — b z—c 

by eliminating y ajid x, and then write out the direction cosijtes as above 
equal to each denominator divided by the square root of the sum of the 
squares of all three. 



NOTES ON SOLID GEOMETRY. jj 

Thus to find the direction cosines of the hne 

y — 7iz-^q 

X— p V — q z 

we^yrltelt ■ ^"^ == — . 

m n 1 

Hence 

m n . I , 

cos a — /-:j— - o , y cos 6 — / . , , ~ cos y— . „ , - 

(39) 

Ex. I. Find the direction cosines of the lines 

3 ~ 4 ~ 2 ^'^^' 

y = 22 + s ) , . . 2x-i-sy-\-62 =z 24 , ^ 
y ■- 30—1 f ^^^ ' 3.r— 4_y+20 r= 10 ^ '^* 

53. To find the cosine of ihe angle heiiveen hvo lines given by the equa- 
tions 

x—a y — h z — c x— a' y—b' z — c 

We have shown (Art. 15) 

cos V = cos a cos a' -{• cos /3 cos /?^+ cos y cos y', 

XT .r LU + MM'4-NN' , , 

Hence cos v — — — . (40) 

■rr ^ ^' i • i r X — VIZ ■]- p ) X =: PI Z ■\- p' 

If the hnes be in the form ^ V , , 

,y — nz^-q \ y — n z-^q 

rvu M m m' + nnj-i 

Then cos V = = — =1. (41) 

V I + ^^^^ + 71' \^ 1 +?n^ + ?i'- 

Ex. I. Find the cosine of the angle between the lines 
x=22 + 6] ^^^ and^^'==' I (2) 



y= 3^—1 ) >' =—2-^2 

Ex^ 2. Find the cosine of the angle between the lines 

x + 2y + sz=5)^,) and -^'--^ + ^"'=='1(2) 
x—ji—z =4) x+j + z —2) 



32 NOTES ON SOLID GEOMETRY. 

These equations may be put in the forms 



X- 


13 


I 




X- 


_3 


1' 


I 






3 . 

I 


--^ 3"_ 2 
4 -3 


(0 


and — 


2 
"5 




2 
3 


~ 2 


rns 


V = 


-5 + 12 - 6 
V26.V38 


VI 


I 












26x38 





=4> (3) 



54. The condition of perpendicularity of two lines given by the 
equations in last article is LL' + MM' + NN'^ o. (42) 

The condition that they shall be parallel (see Art. 15) 

is (LM'-L'M)2 + (LN^-L'N)H(xMN'-M'N7=.o 

or -ir-p= iTFT =1^:77(43). These two conditions when the lines are in the 

X = 7nz-\-p ) X = viz-\-p^ 
forms , V / , ^ 

y z=z nz-\-q \ y = n z-\-q 

become mvi' ^-nn' -\-\ = o, (44) and vi = m\ 7t — 7t' (45) respec- 
tively. 

55. To find the condition of the intersection of two lines 

X = mz -\-p ) . X ■=! VI 'z -\-p' 
V ana ^ ^ 

y =: nz-\-q ) y ^^ n z-\-q 

This is derived by eliminating x, y and z from the four equations. 

Subtracting the third from the first we have o ={ni — m^)z-]-p—p'. 

p p' q — q' 

/. z = -^, — ^— . Similarly from the second and fourth z ——, , 

7n —VI ^ n —n 

and since the lines intersect these two values of are equal. There- 

fore we have , = , . (46) 
m — VI n — n ^ ' 

Ex. Find / so that the lines -^ = ^^ + 3 ) x^zz^i\ ^^^j^ .^_ 
^ y — Z-^l \ y— 2z-\-q ) 

tersect. • 



If the two lines are in the form 

V — l> _ z—c . 

M ~ N ^^^' V "" M' ~ N' 



x—a v — b z—c , '^"~'^'_ y^^' _ -^ "~ '^^ / \ 



NOTES ON SOLID GEOMETRY, 



33 



the elimination of x, y and z can be effected more readily by writ- 
ing (i)=Kand (2) =K'. 

.*. x—a = LK 






X—a — L . 



Similarly h~V = M'K'—MK 
c-c' = N'K'-NK. 

Therefore eliminating K and K' we have 

L, — L, a — a' 
M, -M', d-i>' 
N, -N', c-c' 

or 

(a-a')(NM'-MN') + (5-<5')(LN'-L'N) + {c-c'){h'M -LM') = o. 

(47) 

T/ze Straight Liite under Given Conditions, 

56. The equations of a straight line parallel to one of the coordi- 
nate planes as xy are z ^=^ c, y r=- mx-\-p. 

The equations of a straight line parallel to one of the coordinate 



axes as z^ are 



X = a") 
y = b\ ' 



57. To find the equations of a straight line passing through a given point. 
If (x , y, z') is the point 

-v^ jV" '1/ y 2, Z 

we have seen the equation is — ^ — ^ ivf — ~ — N~ ^^^^ 
or if the equations are in 

the form "" ^ \ then " ~ ) ,; y (40). Hence if the 

y =inz-\-q ) y—y' — n\z—z ) ) 

equations of a straight line contain only two arbitrary constants, all 
the lines obtained by assigning values to these arbitraries pass through 
a single point. 

58. To find the equations of a straight line passing through tivo given 
points {x,y', z') [x\ y\ z') using (48) we' have 

x^'—x v' — v z'—z' 

— =: —- — — — — — ^z — , or dividing? (48) by this to eliminate 

L M N o \t / / 



34 NOTES ON SOLID GEOMETRY, 

L, M^ and N we have 

x—x^ __ y—v _ z—z' . 

If one of the points as {x\ y, z') be at the origin then the equa- 
tions become 

X y z 

-^=y=7- (51) 

59. 7^ find the equation of a straight litie passing through a given 
point {x',y\ z) aiid parallel to a given straight line 

X — a y—h z — c 

From the first condition we must have ^ ^, ~ Ar ~" "n^ 

^ ^. . L M N 

and n-om the second condition -j-^ z=— — r=— y. 



Hence the required equation is 

X — X y—v' z—z' 



. (52) 



L U N 

If the equation passing through the point x\ y', z' be of the form 

x—x' — m'(z—z') ) , , . ,. . -^ ~ mz-\-p 

,; ,/ >- , and the given hue be 
V—y =: 71 {z—z ) ) ^ y— nz-\-q. 

Then 71 = n and ;;/ = ;;/, and the line will be 

^^^r^ = -(^-^') ) (53) 



y—y = ?l(z — z). 

60. To find the equations of a straight line passing through a given 
point x\ y', z' and perpendicular to and inter secti7ig a given right li7ie 

X — a y — h z — c 

I ~ 7n '~ 71 ' 

The required line by the first condition wdll be of the form 

x—x' y—v' z—z' 

where L, M, and N are to be determined by the conditions 
1.1+lslm-V^n = o (Art. 54) 



NOTES ON SOLID GEOMETRY. 35 ■ 

and 

(^-jr')(M;^--N;;2)+(3-y)(N/-L;z) + (^--0')(L;;z--M/)r:zo(Art.55). 

6i. Ex. I. Find the equation of the line joining the points (b, c, a) f / i 

and (C^^) ^^d show that it is perpendicular to the line joining the / hj^^ -^ 
origin and the point midway between these two points ; and that it 

X V 2 

is also perpendicular to the lines x =: v =. z and — =—-=-. 

a b c 

Ex. 2. The straight lines which join the middle points of the 
opposite sides of a tetrahedron all pass through one point. 

Take O one of the vertices as origin and OA, OB, OC as the 
axes of ^, y, z. 

Let M, M', M"be the middle points of BC, AC and OC respec- 
tively, N, N', N" the middle points of the edges OA, OB and AB 
opposite to those respectively. Then to find the equations of the 
lines MN, M'N', M"N". 

We apply the equation —, -== , "' ,, ——7 rr to the points 

^^ ^ ^ x—x' y —y z—z 

(Mj, NO (]\r, N') (M", iN^') respectively. 

Let OA = 2a ; OB = 2b ; OC =r 2c. 

Then M is (0, b, c) and N is (^, 0, 0), 

Hence the equation of MN is 

'y ^ / x 







x—a y 


z 




-a ~ b~ 


c 


Similarly 


the equation of M'N' is 








X y-b 


z 




a -b 


c 


An 


id the 


equation of M"N" is 








X _y __z- 


-c 



b~ -c 



(2) 



(3) 



(i) and (2) give x — — , y =—, z —- and these values satisfy (3) 

Consequently these lines pass through the point (-,-,- 

\ 2 2 2 




36 - ^'OTES ON SOLID GEOMETRY, 



Straight Line and Plane, 

62. To find the conditio7is that a line shall be perpendicular to a plane 
given by its equation. 

If the plane be of the form x cos a-\-y cos /? + cos y =z p (i) 
we know that cos a, cos /?, cos y are the direction cosines of the 
perpendicular from the origin on the plane. 

And the equation of this perpendicular will be 

X y z 



cos a cos 16 cos y ' 

If any plane Ax + By + C0 = D be parallel to the plane ( i ) we must 
have 

A _ B _ C 

cos a ~ cosp ~~ cos y 

and if the line — ^ — ~ ' AT ~ N ' ^^ Parallel to the line 

X j; z ^ 

■ = ^= , we must have 

cos a cos p cos y 

L ]\I N 



cos^ cos/y cosy * 

Hence the conditions that the line — ^^ — = ' =-<¥- shall be 

L, 1\1 IN 

perpendicular to the plane Ax + Bv + Cz = D will be 
ABC 



L ~ M ~ N • 



(54) 



Tr.u r "u • .-L r X = ??IZ-}-p ) . . X—p J' — O Z 

If the hne be m the form ^ \ we write it- — - — - — ^ = - . 

y :=i 7iz-\-q ) VI p I 

And the conditions are ^ — = — == — or -^ ^Y (sO 
;;/ ;/ i B = ?iC ) ^-^"^^ 

The equation of a line passing through the point .r',y, <&' and 
perpendicular to the plane Ax + By + Cz = D will then be 

x—x'_ y—y' _ z—z' 
"~A~'^'~B~~~"C~* 



NOTES ON SOLID GEOMETRY. 37 

or in the unsymmetrical form 

Ex. Find the equation of a line passing through the point (i, 2, 
3) and perpendicular to the plane ^x-{-2y — 40 = 5. 

6^, To find the condiiion that a straight line shall he parallel to a 

given plane. Let the plane be Aj; + Bv4-C0 = D and the line of 

» ^ x—a y—b z—c 

the form ^-=-3^^^^. 

Now if this line is parallel to the plane it will be perpendicular to 
the normal to the plane. Hence the required condition will be 

AL + BM + CN = o. (56) 

64. To find the conditions that a straight line shall coincide with a given 
plane Ax + By + Cz = D. 

x—a y—h z — c 



1°. Let the line be of the form 



M ~ N 



The line must fulfil the condition ( ) above of parallelism above, 
AL + BM + CN =::o. And also any point on the line as {a^ 3, c) 
must satisfy the equation of the plane. Hence we must have the 
additional condition A.a-\-Bb-\-Cc—'D = o. (57) 

2°. Let the equations of the line be of the form x=mz ■\-p \ 

y=i7iz-{-q ) . Sub- 
stituting these values of x and y in the equation of the plane, we 
have 

A(m+/)+B(;/0 + ^) + C0 = D, 

whence 7.^=^— -^ And for coincidence this value of Z must 

Km^^n-\-C 

be indeterminate, and therefore A/ + B;7— D = o ) (rg) 

Aw + B;^ + Cr=oj are the condi- 
tions of coincidence. 

Note. This last method is a general one of determining the con- 
ditions coincidence of a straight line and any surface given by its 
equation. That is substitute x andjj/ of the line in the equation of 
the surface and since the z in the resulting equation must be inde- 
4 



38 



AZOTES ON SOLID GEOMETRY, 



terminate if there be coincidence we treat this equation as an iden- 
tity and make the coefficients of the different powers of z separately 
equal to zero. 

65. To find the expression for the length of the perpeiidictilar VT) from 

any point P(x', y',z') on a straight line AB given by its equation, 

o T 1- • 1 x — a v—h z — c 1 , 1 ,. 

I . Let hne be ■=- -= where a, b, c are the coordi- 

cos a cos p cos y 

nates of any point A on the line. Now PD^ ~ PA^— ADl 

But PA^ ■= [^x'-aY^\-{y -bY-V{z-cf and AD being the projec- 
tion of PA on AB, we have 

AD =: {x—a) cos a-\-i^y—b) cos ^^-{z—b) cos y. 

Hence 
YVi'^^x' -a)''-\-{y -bY y[z' -cY-{{x-a) cos a^{y-b) cos y5 

+ (0-^) cos;/). '(59) 

2°. If the given line be of the form ^ 

x—a y — b z — c 

Then 

A 

cos a: = — ===== ; etc., etc. 

And therefore PD'^ 

3°. If the given line be x := mz-\-p 
y ^=. jiz^q 
Then PD^ 

66. To fijid the expression for the shortest distance between two straight 
lines given by their equations. 

This shortest distance is a straight line AB perpendicular to both 
the given lines PB and SR. 
Let the given equations 

X — a v—b z — c , x—a y—b' z — c' . ^ , 

=:^ -=— — and 7—- — 777= 7 and 6 = the 

cos a cos p cos y cos a cos p cos y 

angle between the lines. 

And L, M, N the direction cosines of the perpendicular AB. 



NOTES ON SOLID GEOMETRY. 3g 

Then we must have 

L cos ^ + M cos /? -f N cos y = o ) 
L cos a -h M cos /5' + N cos y' = o. ) 
Whence 

L M - 

cos p cos ^' — COS j3^ cos y cos a cos y' — cos a cos y ~" 

N 



cos a cos //'—cos a! cos ^ ' 



VL'+ lAP+N' 



.< 



(cos /3 COS y' — cos /3^ COS y;2_j_(cos a COS y' —COS a' COS y)2-|-(c0S a COS ^' — COS a' COS ]8)2 ; 

I 



sin d 
(Art. 15). 

Now let P be the point [a, 5, c) on the line PB and Q be the point 
{a\ b\ c) on the line SR. Then as the projection of PQ on AB is 
AB itself, we have 

AB =: (^-^')L + (^-^')M + {^-^')N =: 

(62) 

(a— aO(COS^COSv'— C0«7C0S^')+(^— ^')(COSaCOSY'— C0Sa'C0S7)+(C— C')(COSaCOSj8'— COSa'COSjS) 

"" sin 6, 

If the given lines are expressed in other forms we can find cos a, 
cos /?, etc. from the given equations and substitute them in {^^2), 



CHAPTER V. 



TRANSFORMATION OF COORDINATES. 

67. To transform^ to parallel axes through a new origin the coordi- 
nates of which referred to the old axes are a, b, c. 

Let OA =: X, AN ^=i y, FN = s be the coordinates of/ referred to 
the origin O and the axes Ox, Oy and Oz. Also let O' be the new 
origin, and OA' 1= a, A'N' — b, N'O' == r be its coordinates and let 
O'H 1= x\ HK z=zy' and PK = z be the coordinates of P referred to 
O' as origin and axes parallel to the original axes. 

Then x = OA = OA' + A'H. 

Similarly x := a^x' \ 

and z ^ c-\-z' j 

Substituting these values in the equation of a surface we obtain the 
equation referred to the new origin and axes. 

68. To pass from a rectangular system to another system the oj'igin 
remaining the sa?ne. 

Let Ox, Oy, Oz be the old axes at right angles to each other Ox', 
Oy\ Oz the new axes inclined to each other at any angle 

OM = X, MN =y, NP = 
O^V :=x\ M'N' =y, NT = z\ 

Now the projection of the broken line OM'-f MN' + N'P on the 
axis Ox is equal to the projection OM of the radius vector OP on 
Ox. Let cos a, cos (3, cos y be the cosines of the angles which the 
new axes make with the axis Ox ; then 

X = x' cos a-\-y' cos l3-\-z' cos y, 

40 



NOTES ON SOLID GEOMETRY, 



41 



If COS a\ cos /?', cos y' be the cosines of angles which the new 
axes make with the axis Oy, and cos a' , cos /5", cos y'\ the cosines 
of the angles which they make with Oz, we shall have similar values 
for V and z. Hence the three equations of transformation are 

X =: X' cos ^ +y cos /3 -{-Z COS y \ 

y — X cos a +y cos f5' ^-z' cos y' V (64) 
z — x' co^ a" -{-y co^ 13" -{-z' co^y". ) 

We have of course 

cos' a + cos' /? + COS^ y r=z l \ 

cos' a' +cos'/5' +cos' / = i > (B) 

cos' <^''+C0S'/5" + C0S' ;^" r= I. ) 

For the angles A, fi, 1^ between the new axes ofy and z\ of 0' and 
x\ of x' and y respectively we have 

cos A = cos a' cos a"+ cos (3^ cos ^" +cos ;k' cos y" \ 
cos yu = COS a" co?> a + cos /5"cos y5 +cos ;k"cos / > (C) 
cos r = cos a cos <x' + cos ^ cos /?' +cos y cos }/'. ) 

69. To pass from one system of rectangular coordinates to another also 
rectangular. 

The formulae in this case are the same as those in the last with the 
exception that since the new axes are also rectangular cos A=o, cos }x 
= o, cos V =0 and formulae (C) give 

cos a^ cos (^'' + cos ft' cos /5" + cos y cos 7''= o \ 
cos a" cos a + cos /^"cos /? +cos ;k' cos ^ = o V (D) 
. cos a cos a:' + cos /? cos y^' +cos y cos ^' == o. ) 

Since between the nine quantities there are six equations of con- 
ditions, (B) and (D) there are only three of the quantities, cos of, 
cos f3, etc., independent. 

70. In changing from rectangular axes to rectangular, there is 
another set of equations of condition among the quantities, cos a, 
cos ^, etc., equivalent to the preceding which result from the fact 
that the new axes are rectangular. For a, a', a!' being the angles 
made by the old axis of x with the new rectangular axes, etc., we 
must have 

cos' a + cos' a! + cos' a" zn \ \ 
cos' /^ + cos'/5' + cos' (3" ^\\ (E) 
cos' y + cos' y + cos' y" z=z i ) 



42 NOTES OX SOLID GEOMETRY. 

COS a cos /?4-cos a cosy5'+cos a^ cos /5" = o ] 

cos a cos ;k + cos a' cos ;k' + cos flf'' cos y'z=.o \ (F) 

cos ji cos ;^ + cos yS' cos 7' + cos /5" cos /" =: o ) 

and the new coordinates expressed in terms of the old are 

X = X cos a -\-y cos ft -\-z cos y ) 

y = a: cos a -\-_ycos /3' +s:cos y' > (F) 

z' r= X COS a" +^ COS/5'' -f 2: COS /" ) 

71. In the Study of surfaces by sections made by planes it is often 
necessary to transform the coordinates in space to coordinates in 
the cutting plane. To do this we must fix the plane with reference 
to the old coordinate planes. Let the equation of the plane be given 
as z =Ax-hBy, Then the angle 6 which this makes with the plane 

xy is determined by the equation cos 0— — = and the 

angle cp which it traces on that plane makes with the axis of x 

by the equation tan cp = — —, the trace being A.v + B;^ = o. 

Let x'Oy be the given plane, cutting the plane xy in the line 
Ox' which take for the axis of x' and let Oy' a line perpendicular 
to it in the given plane be the axis of ji^' and OR = x\ RM =y 
the coordinates of any point M in the plane referred to the axes O.v', 
Oy ; also let OQ = x, OP =j;, PM = ^ be the coordinates of M 
referred to the old axes O.v, O;', O-s:. Then the angle ]\IRP = d 
and XOX' = cp. 

Then PR =/ cos 6, V^l =zy sin 6, 

OQ =: OR cos ^4-RP sin cp, QP= OR sin cp-RT cos cp. 

/. z =y sin ) 

X = x' cos ^+y cos s'm cp y (65) 
J' = x' sin cp—y cos 6 cos cp ) 

And if these values be substituted in the equation of any surface 
Y{x, y, z) ~ o the result will be a relation bet^veen x andy, coor- 
dinates of the curve cut from the surface by the plane. 

72. If the cutting plane contain one of the coordinate axes, the 
formulae are simplified and in many cases sufficiently general. 

Let X'OY be the cutting plane containing the axis ofj-; O^' its trace 
in the plane zx the axis of x', PM=a;', OM —y. the coordinates of 



NOTES ON SOLID GEOMETRY. 



43 



any point P in the section, ON =3 x, NQ ^=.y, QP = z the coordi- 
nates of P referred to the old axes. Then angle PMQ = d, and 
MQ ■=. X cos d, PQ = X sin ft . *. The formulae of transforma- 
tion are 

X = x' cos 6 ) 

y=y' \ (66) 

z ^=z x' sm 6 ) 

That is, we have only to make x := x' cos 6, z =: x^ sin d, y ^y in 
the equation of any surface, in order to find the equation of the sec- 
tion of this surface by a plane, containing the axis of^and making an 
angle d with the plane xy. 



CHAPTER VL 

THE SPHERE., 

73. To fijid the equation of the sphere, 

1°. In rectangular coordinates. 

Let a, b^ c be coordinates of the Centre, and Radius = R. 

The equation is then (Art. i i)(.r — ^)' + (jV — ^)' + (2 — 0'=R- (67) 
or if the origin be at the centre 

x'-\-f^z^^V^\ (68) 

2°. In oblique coordinates. 

Let A, //, V be the angles of the axes then the equation is (Art. 16) 

2[x—a)(z—c) cos }x-\-2^y—h)(z—c) cos t^ z= R^ (69) 

or if the origin be at the centre 

^^ + j^ + 2:^ + 2a7cos \-\-2xz cos }x-\-2yz cos V = R^ (70) 

3°. In polar coordinates 

Let r', a, /3 be the polar coordinates of the centre then the equa- 
tion is 

r^ + r- — 2rr'(cos 6 cos ^4-sin 9 sin a cos {cp — /3)) = R\ (71) 

If the pole be at the origin and the centre on the axis of 0, the 
equation is 

Y^= 2R cos e, (72) 

Since that is the equation of the generating circle in any one of its 
positions. 

44 



NOTES ON SOLID GEOMETRY. 45 

74. The Sphere under conditions (coordinates rectangular). 
The equation ((i^]) may be written 

or x'-i-/+2'-{-'Dx-i-'Ey + F2 + G = o, {js) 

Auxd since this equation contains /bur arbitrary constants, the 
sphere may be made to fulfil /bur conditions (which are compati- 
ble) and no more. Four given conditions give four equations for 
determining the constants D, E, F, G, and with these determined 
we know the radius and centre of the sphere, for we have only by 
completing the squares to throw the equation ( ) into the form 

to see that the centre is ( , , ) and the radius is 

\ 2 2 2/ 

J32 £2 p2 

- + — + G. 

4 4 4 

1°. The equation of a sphere passing through a given point A, e, f, is 

x^^y''-^z'^'Y){x-d)-\-Y.{y-e)-\-Y{z-f)-d'-e'^f':=^o, (74) 

If the given point be the origin the equation is 

^'+y + 32 + Dj^ + E^ + F0r= o. (75) 

2°. The equation of a sphere cutting the axis of z at distances c and d 
from the origin is 

x^-\-f^{z-c){z-c)^T)x^Yy^ o (76) for"^^^ ^ 

must give two values for 0, c and c, and this equation fulfils that 
condition. 

3°. The equation of a sphere touching the axis of z at a distance cfrom 
the origin is 

x^-^y'^-\-{z—cY\-^x^Y.y — o (jj) for this gives two coinci- 
dent values of = ^ when " ~ 



46 * NOTES ON SOLID GEOMETRY, 

4°. The equation of a sphere touching all three axes at distance 2ifrom 
origin. 

To meet these conditions the equation must be of such a form as 

to give equal roots for z when _ > the same equal roots for y 

when ' \ and the same equal roots for x when -^ "" v . Let 

Z — O \ ^ z — o \ 

the distance of points of contact from origin be a, then the equation 

will be 

x'±2ax^y'±2ay-\-z''±2az-^a^ =.Q (78) 

as this fulfils the above conditions. 

5°. The equation of a sphere passing through the origin and having its 
centre on the axis ofx\s> 

x'-\-f-\-z'^2Rx. (79) 

6°. The equation of a sphere tangent to the plane xy at the point (a, b) 
is 

{x-aY^{y-hy--\-z--\-Yz^o (80) 

for then 0=0 gives x^=^a, andj'=(^, a point {a, b) in the plane xy, 

75. Interpretation of the expression 

(^x-aY^{y-hY^(z-cy-^\ (i) 

1°. Let (jc, ji', 2:) be the coordinates of a point P without the 
sphere whose centre O is (a, h, <:) and radius == R and let P^I be 
tangent to this sphere at the point M. Then P]\P =: OP^ — OMl 

Now QY^^^x-aY-^{y-hY-\-(z-cY 

and hence PM^ = {^x-aY-^^y-Vf -V[z-'cY-'^\ 

Therefore the expression (i) is the square of the tangent from the 
point P to the sphere. 

2". Let Y{x,y,z) be a point within the sphere. Join OP and 
erect a perpendicular PM to OP meeting the sphere in i\I, and join 
OM. 

Then P]\Pi=OM"^-OP = ^' -{{x-aY^iy-hY -Viz-c)') 



NOTES ON SOLID GEOMETRY, ^y 

That is the expression (i) becomes negative and represents the 
square of the half chord through P perpendicular to the radius 
through P. 

76. Radical plane of two spheres, 

Def. The radical plane of two spheres is the plane the tangents 
drawn from any point of which to tlje two spheres are equal. 
If the equations of the two spheres are 

the equation of their radical plane is 

{x-aY^{y^hY^{z-cy-^^'-{{x-ay^{y-^hJ-^{z-cJ^^') 

— o 
For this expresses (Art 75) that the squares of the tangents from 
point {x, y, z) to the two spheres are equal^ and moreover it is an 
equation of the first degree in x, y and z and therefore the equation 
of a plane. If the spheres intersect their radical plane is their 
plane of intersection. It may be easily proved that the radical 
plane of two spheres is perpendicular to the line joining their cen- 
tres. '■■ - 

J J, The six radical planes of four spheres intersect in a common 
point. 

- Let S = o, S' zn o ; S'' = o ; S'"= o be the equations of the four 
spheres. Then the equations of their radical planes are 

S-S" ^o S' -S"'=:o 

S-S"'z=o S"-S'" = o 

These may be added, so as to vanish simultaneously and therefore 
the planes intersect in a common point. This point of intersection 
of the six radical planes is called the radical centre of the four 
spheres. 

78. Examples : 

1°. Find the centres and radii respectively of the spheres 

^^-f-y + 2^— 2^ + 3;'— 52: = 0. 



48 A'OTES ON SOLID GEOMETRY, 

5jc^ + 5^4- 52:'-- 1 2:^+20); +240— 40 = o. 

x^-\-y^ + z^ = 320. 

2°. Find the equation of a sphere passing through the origin and 
the points i, 2, 3, ~i, 4, 5> 3. o^ i- 



CHAPTER VIL 

CYLINDERS, CONES, AND SURFACES OF REVOLUTION. 

79. Cylinders. Def. A cylinder is a surface generated by the motion 
of a straight line which always intersects a given plane curve, and is 
always parallel to a fixed straight line. The moving straight line is 
called the generator; the plane curve which it always intersects is 

called the directrix ox guiding curve, the fixed straight line the axis, 

•* 

80. To find the general equation of a cylinder. 
Let m, n, i be the direction cosines of the axis. 

And let 3^ ^ \ (0 ^^ ^^ equations of the generator in 

which 7n and n are constant since the generator remains parallel to 
the axis. For convenience take the guiding curve in the plane xy, 

its equations will then be ^^^^~~ ' (2) Now making = o in 

2=0 ) ^ 

the equations (i) we obtain x—py—q for the point in which the 

generator pierces the guiding curve F(:v, y) in the plane xy. 

Hence we have F(/, q) — o, (3) and eliminating the arbitraries/ 

and q between (i) and (3) we obtain 

Y{x—mz,y—nz)^:^o (82) 

the general equation of cylinders. 

If the cylinder be a right cylinder with its guiding curve in the 
plane xy and the axis of 2 for its axis, then in equation (82) m z= o, 
and n=^Q, and the required equation of the cylinder is 

Y{x,y) = o, (83) 



NOTES ON SOLID GEOMETRY, ^g 

8i. Cylinders of second order. We shall confine ourselves to cylin- 
ders whose equations are of the second degree. 

1°, To find the equation of the oblique cylinder with circular base. 

Here Y(x,y) = x^ +/ — R^ — o. Hence Y(x—mz^ y—nz) — o 
%\\^%(x—mzY-\-{y'-nzf—^^o (84) the required equation. 

2°. To find the equation of the right cylinder with circular base. If 
the axis be the axis of z, the equation is F(^) = o that is 

3°. To find the oblique cylinder with elliptical base. Let the guiding 

X^ 1/2 

curve in plane ^y be - +^ = i. 
^ -^ a^ b' 



b'' 



X V 

Then ¥{x,y) = ^+^—1=0 and the equation is 



(x — mzY (y — nzY 
a' "^ b' ^ 



4°. The equation of the right cylinder with elliptical base whose 



.^2 ^,2 



X V 

axis is the axis of z is F{Xy y) = o, that is, — + ^=: i. 

5°. The equation of the right parabolic cylinder whose axis is the 
axis of is y^^4dx = o or^ = 4dx. 

82. Cones. 

Def. A cone is a surface generated by a straight line which passes 
through a fixed point and always intersects a given plane curve. 
The fixed point is called the vertex, the moving line the generator, 
and the given plane curve the directrix or guiding curve, 

^'^, To find the general equation of a cone. 

Let the coordinates of the vertex be {a, b, c) the equation of the 

x—a y~b z—c , . , , 1 ,. . . , 
generator = = (i) and take the directrix in the 

plane (jcy)— its equation being then _ j- (2). Now if 

eliminate the arbitraries m and n between the equations (i) and (2) 

the result will be the equation to the cone, the locus of the right 

line (i). 

^ 2ZI a mc ) 

Making = o in (i) the values of x andy, namely, _ , >• 

which result will be the coordinates of the point in which the gene- 
5 



we 



50 



NOTES ON SOLID GEOMETRY. 



rator meets the plane xy and these will consequently satisfy F(^, y) 
= o the equation of the directrix. We have therefore ' 

Y{a—mc, b—nc) = o (3). But from (i) m =- , n =— , 

z c z — c 

and therefore (3) becomes 



or 



\ Z — C Z — C 



the general equation of cones. If vertex be on axis of z, then a = o 
and d =zo and equation (85) becomes F( — — -, —j =z o. (86) 

84. Cone with vertex at origin. 

If the vertex of the cone is at the origin and the directrix in a 
plane parallel to the plane xy, and at a distance c from it then the 

X V z 

equation of the generatrix will be — =z'—=z — , (i) the vertex (o, o, o) 

m n I ^ 

and the directrix will be ^ -^^^ ~ I (2) 



z ■= c 
To find the point in which the generator meets the directrix we 



X ^^ nic ) 

make0r^<:in (i). We thus get ]■ 

^ ^ ° y :=zi nc ) 



X V 

Hence we have Y{niCj nc) =0, but /?/ = — , and « =— from (i). 

Therefore 

F(f,f)=o m 

is the equation required. 

The equation (87) is a homogeneous equation in x,y and z, 

85. Cones 0/ second degree. 1°. The eqiiaiion of an oblique cone with 
circular base. 

The equation of the directrix is Y{x, y) = x^-\-y^— R^ = o. 
Hence 

\ Z — C Z — C I \ Z — C I \ Z — C J 



NOTES ON SOLID GEOMETRY, 5 1 

or {az'-cxY-\-(hz-cyY^^\z-c)\ (88) 

2"^. To find the equation of a right cone with circular base, the axis of 
z being the axis of the cone and vertex being (o, o, c). The equation 

of the directrix IS ^ -^ ' -^ 

z=^ c. 

^ ^[—cx ^cy\ . c^x^ cY ^, 

Hence F( , — ~)==o is -, ^4- , ' ,, -R' 

\z—c z—c) 






or Ar^4-y=-j(0— r)^ (89). This is a cone of revolution about the 
axis of <sr. 



3°. The equation of a right cone with vertex at the origin and circular, 
elliptical, or hypej'bolic bases. 

The equations of the circular base (directrix) are 



}• 



F(jt:,j^')=;vH/-R'=o 

2:— c 

Hence 

f ex cy\ c^x^ cY R^ 

Ff — , ^\=zo gives -^H — ^ — R^=:ooTx'^'{-y=~.z\ (90) 

The equations of the elliptical and hyperbolic directrices are 
^ +^. - I ^ o f and -,~- - 1 =. o f respectively. 

Z = 0) Z =: o) 

Hence the elliptical and hyperbolic cones are 

c'x' cY x' f z' . , 

-^ + ^-^.-i-oor-+--z=- (91) 

c'x'' cY x" y r 

86. Surfaces of Revolution. 

To find the general equation of a surface generated by the rroolution of 
a plane curve generator about the axis of z. 

Let SP=:r be an ordinate of the point P to the axis of z of the 



52 



NOTES ON SOLID GEOMETRY, 



plane curve and OM = x, MN =jk, NP = z the coordinates of P. 
Then SP^ = ON^ = O^P + i\IN^ or r' - x'-^y'. 

That is, the distance from any point of revolving curve (gen- 
erator) from the axis of z is r=L^x^-\-y^ (i). But r being an ordi- 
nate of the generating curve to the axis of z we must have by the 
equation of the curve in any position r = F(0) (2). Therefore 
eliminating the arbitrary r between (i) and (2) we have 



V^^' = F(2) (93) 

the required equation of surfaces of revolution about axis of^r. 
If the curve revolved about the axis of ;v the equation is 



Vy+? = F(.v). (94) 

87. Surfaces of revolutmi of secojid 07'dei\ 

1°. Equation of Cylinder of revolution about the axis of z. The 
equation of the revolving line is = a. 



.*. V-^'+y^ = F(2^) gives x-+y =: a^, 

2°. Equation of a Cone of revolution about the axis of z, vertex at 
(o, o, c). The equation of the generating line is r ^ 7n[z—c), 

Hence or^+y = ??i^{z—cy (95) the required equation where m is 
the tangent of the angle made by side of cone with axis of 0. 

3°. Equation of the Sphere. The equation of the generating curve 
is r^ + 'S:^ ^=z a^ or r z=z^fa^—z^. 

Hence V^t^M-y = "f a^— z^ or x'^ +y^-\-z^ — al 

4^. Equation of the Surface generated by the revolution of an ellipse 

about its conjugate axis, 

r^ z^ a^ 

The generator is -. + 7^ = i or r^ = i^(3^— 2:^). 
a 

Hence the equation of the surface is 

oc^ -l- v^ z^ 
or _^ + _=i. (96) 

This is one of the ellipsoids of revolution called the oblate spheroid. 



NOTES ON SOLID GEOMETRY. 



53 



5°. Equation of the Ellipsoid generated by the revolution of an ellipse 
about its transverse axis the {^Prolate spheroid). 

Take the axis o{ x as the axis of revolution. Then the equation 

x^ r^ 
of the generator is -^ + -5 = i 
a 0' 

32 

or r^ = — aC'^'^— ^^). Hence V^'^ + 'S:^ — Y{x) gives 

^ + ^=1 (97) 
the required equation. 

88. Hyperboloids of revolution. Definitions. When the Hyperbola 
revolves about its conjugate axis it generates the Hyperboloid of revo- 
lution of one sheet. When it revolves about the transverse axis it gen- 
erates the hyperboloid of revolution of two sheets. 

1°. Equation of the Hyperboloid of one sheet. Let the axis of z be 
the conjugate axis then — — — = i or r^ =: — (2;^-f<5^). Hence 

2°. The equation of the Hyper'boloid of revolution of two sheets. Take 
the axis of x as the axis of revolution. Then the equation of the 

x'^ r- <5- 

generator is — — -^ — i or r^ — — (:v^— a^). 

Hence for the equation of the surface we have * 

^--^=1. (99) 

89. Equation of the Paraboloid of revolution about the axis ofx. 
The equation of the generator is r^ = ^dx. 

Hence the equation of the Surface isj'^ + 2:^ = Adx. (100) 

5* 



CHAPTER VIII. 
ELLIPSOIDS, HYPERBOLOIDS, AND PARABOLOIDS. 

89. To find the equation to the surface of an Ellipsoid. 

Def. This surface is generated by a variable ellipse which always 
moves parallel to a fixed plane and changes so that its vertices lie on 
two fixed ellipses whose planes are perpendicular to each other and to 
the plane of the moving ellipse, and which have one axis in common. 

Let BC, CA be quadrants of the given fixed ellipses traced in the 
r* planes j^^^, zx \ OB = c their common semi-axis along the axis of 0, 
OA = a (on the axis of ^), and OB = b (on the axis oi y) the other 
semi-axes ; QPR a quadrant of the variable generating ellipse in any 
position, having its centre in OC and two of its vertices in the ellipses 
AC, BC, so that the ordinates QN, RN are its semi-axes ; also let 
ON — z, NM = x^ MP —y be the coordinates of any point P in it: 

Then -^ -I- ^^ =z i. And since Q is on the ellipse AC we 

. QN^ z' ^ ., , RN^ 2' 

h^ve -^=i__ Similarly -^= I--. 

Hence eliminating RN^ and QN^ we have 
x^ f 



"■(-7:) '■(-;■.) 



X^ 1/2 2^ 

the equation to the surface. 

90. To deternwie the form of the ellipsoid from its eqiiaiio7i. Since in 

x^ y^ z^ 
the equation — ^-l--^ + ^ = i, x can only receive values between a 

54 



NOTES ON SOLID GEOMETRY. 55 

and — a,y between h and — b, and z between c and — r, the surface 
is limited in all directions. 

If we put 0=0 we obtain -^ + ^ — i, for the equation to the 



a 



b^ 



trace on xy, which is therefore the ellipse AB. 

X^ 02 

If weputj^== owe have —^-{.—^ —i^ the ellipse AC. 



a* 

»,2 



If we put :v = o we have '-—-^—-=.1. or the ellipse BC. 

c 

These three sections by the coordinate planes are called the princi- 
pal sections, and their semi-axes a, h, c, are the semi-axes of the ellip- 
soid ; and their vertices the vertices of the ellipsoid, of which it has six. 

If we make z^=^h we have 



d^'^ b^~~^ c"' 

the equations of any section parallel to xy, which is an ellipse similar 
to AB, since its axes are in the ratio oi a to b, whatever be the value 
of ^, and which becomes imaginary when h ^ c. In the same man- 
ner all sections parallel to xz^ and yz are ellipses respectively similar 
to AC and BC. The whole surface consists of eight portions pre- 
cisely similar and equal to that represented in the figure. 

x'^ -\- y^ z^ 
Cor, If3=athe ellipsoid becomes ^ h-y- = i the ellip- 
soid of revolution about the axis of z. Art. (87), all the sections of 
which by planes parallel to_>';^, are circles. Hence the spheroids may 
be generated by a variable circle moving as the variable ellipse, in 
Def. Art. (89). 

^ 91. To find the equation to the hyperboloid of one sheet. 

Definition. This surface is generated by a variable ellipse, which 
moves parallel to a fixed plane, and changes so that its vertices rest 
on two fixed hyperbolas, whose planes are perpendicular to each 
other, and to the plane of the moving ellipse, the two hyperbolas 
having a common conjugate axis coincident with the intersection of 
their planes. 

Let AQ and BR be the given hyperbolas traced in the planes zx.yz ; 
OC = c their common semi-conjugate axis coinciding with the axis 
of z ; OA = a, OB = b the semi-transverse axes ; QPR the generating 
ellipse in any position having its plane parallel to xy^ its centre in 



56 NOTES ON SOLID GEOMETRY. 

OC, and its vertices in the hyperbolas AQ, BR, so that the ordinates 
NQ, NR, are its semi-axes. Also, let MN = ^, MP =7, ON = z, 
be the coordinates of any point P in the generating ellipse ; then the 
ellipse PQR gives 



NQ^ ' NR' 
Also from hyperbola AQ — ^ ^ — ^« 



c 



NR^ 0- 
And from hyperbola BR ~f^-""» — = i« 



Hence^ 



•S^ \ r„/-S^ 



^'^ r 



+-.-/-.=. 



or ""?"+ ^ — ~-r ~ I (^02) the equation to the surface. 

92. To determine the form of the hyperloloid of one sheet from its 
equation. 

Since the equation (102) admits values of x, y and z positive and 

negative however large, the surface is extended indefinitely on all 

sides of the origin. If we put = we obtain 

x^ y^ 

— ^ + ^ = I for the trace on ^r)^ w^hich is the ellipse AB. Similarly 

x^ z^ 
the sections by the planes xz and ^'2: are respectively -^ ^~ ^ ^^^ 

y z 
hyperbola AQ, and^ — — = i the hyperbola BR. The ellipse AB 

and the hyperbolas AQ and BR are the principal sections. The sections 
parallel to xy are all ellipses similar to and greater than AB. The 
sections parallel to xz and j^ are hyperbolas similar to the principal 
sections. 

The -semi-axes a and h are called the r^^/ semi-axes of the surface 
and c the imaginary semi-axis, since a: = o and y ^=.0 give z = 
±<:\/— I. The extremities of the real axes are called the 
vertices of the surface. The surface is continuous and hence is 
called the hyperboloid of one sheet. The hollow space in the inte- 
rior of the volume of this hyperboloid of which the ellipse AB is the 
smallest section has the shape of an elliptical dice-box. 



NOT£S ON SOLID GEOMETRY. 



57 



Cor. If 3 = ^ the equation becomes ~ ^= i that of the 

hyperboloid of revolution of one sheet. Its sections parallel to xy 
are all circles. 

93. To fifid the equation to the hypei'holoid of two sheets. 
Definition. This surface is generated by a variable ellipse which 
moves parallel to itself, with its axes on two fixed planes at right an- 
gles to each other and to the plane of the generating ellipse ^nd ver- 
tices in two hyperbolas in those planes having a common transverse 
axis. 

Let AQ and AR be the given hyperbolas traced in the planes zx, 
xy, OA = a their common semi-transverse axis along the axis of ^, 
OB = h OC = c the semi-conjugate axes along the axes ofj' and z ; 
QPR the generating ellipse in any position having its plane parallel 
Xoyz, its centre in Ox, and its vertices AQ, AR so that the ordinates 
'QN, RN are its semi-axes. Let ON — x, MN —y, MP = be the 
coordinates of any point P in the ellipse. 

v'^ z" 

Then -4-- + ,4. - I 



also from hyperbola AQ 



RN^ ' QiV 

QN- x^ 



RN^ jv^ 
and from hyperbola AR — — ^ 



U" 



a 



Hence jr~, — i r + 



X' 



— I 



r-2 



-^.--TT-^r-^ (103) 

the equation to the surface. 

94. To determifie the form of the hyperboloid of two sheets from its equa- 
tion. 

The equation shows that all values of .r between -\-a and —a give 
imaginary results, therefore no part of the surface can be situated be- 
tween two planes parallel \.o yz through A and A' the vertices of the 
common transverse axis ; but the equation can be satisfied by values 



58 NOTES ON SOLID GEOMETRY, 

oi X, y, Zj indefinitely great, therefore there is no limit to the distance 
to which the surface may extend on both sides of the centre. 

If we make x^=^o we have -r^ + — ^ = — i an imaginary curve 

for the principal section by the plane j^'2:. For x ^=i ± h and h'^ a 

v^ z^ h^ , 

we have ~ -i — — = —^ i which represents similar ellipses. The 

principal sections by the planes xy and zx are AR and AQ, respec- 
tively. For- the sections parallel to xy and putting 2: = ± / we 

x" y ^ 

have — ^ -j^ = I + —j^ a hyperbola similar to AR with its vertices 

in AQ and the opposite branch of that hyperbola and conjugate axis 
parallel to Oy, In the same way the sections parallel to zx are hy- 
perbolas similar to AQ with vertices in AR and its opposite branch 
and conjugate axes parallel to Oj, za is the real axis of the surface 
and its vertices the vertices of the surface. The axes 2a, 2b and 2c 
are the imaginary axes of the surface as it cuts neither J^^ nor z. The 
whole surface consists of two indefinitely extended sheets perfectly 
similar and equal, separated by an interval. Hence its name. 

x^ y^ ^ Z' 

Cor. If d =: c the equation becomes ~ — - — = i the equa- 

a" r 

tion to the hyperboloid of revolution about its transverse axis. 

95. Asymptotic cones to the two hyper boloids, 

1°. The hyperboloid 0/ one sheet has an interior asymptotic cone, 

X^ V* z^ 

Putting its equation — y- + ^7^ Y" — ^ (^) in the form 

x^ y^ z^ / c'^ \ 

—j--\-^z=L~^{i-\ ^ j. (2) Now when z is very great 

—J- is very small, and hence the limiting form of (2) for increased 

z 

without limit is 

X^ y2 ^2 

~~Y + '-jr ~ "T (3) t^^ equation of an elliptical cone having 

its vertex at the origin and its elliptical section parallel to xy. 

Moreover, this elliptical section is always within the corresponding 



NOTES ON SOLID GEOMETRY. 



59 



section of the surface by the same plane. For putting z — ±,h\Yi 
(i) and (2) respectively we have 

^—K- + 4^ = I H — TT for the section of the surface 



x^ y^ k^ 

— ^ + 4^- = -^ for the section of the cone, 

a^ b^ r 

This cone is asymptotic to the hyperbola. 

2°. Tht hyperholoid of two sheets has an exterior asymptotic cone. 

x^ V^ z^ 
Putting the equation — ^ ~-^ ^ =1 (i) under the form 

x^ y^ ^ 

—Y = ^T + '~yI ^ + — ~2 — ^~ — 2" \ w^ ^^ve as a limiting form of 



^ / 



y 



+ -^ 



b' r 
this equation when^ and z increase without limit, 

— ^ rr 4— -I (2) an elliptical cone with vertex at the origin 

a c 

and with an elliptical section parallel to the plane yz. Moreover, 
this elliptical section is greater than the corresponding section of. 
the surface by the same plane. For putting x:=i ±h in (i) and (2) 

1'^ z^ h^ 

respectively we have ^ H ^ = —^ — i 

^ ¥ c^ a^ 

f z' _ h' 

This cone is asymptotic to both branches of the hyperboloid. 

96. To find the equation to the elliptic paraboloid. 

Definition. This surface is generated by the motion of a parabola 
whose vertex lies on a fixed parabola, the planes of the two parabolas 
being perpendicular to each other, their axes parallel and their con- 
cavities turned in the same direction. 

Let OR be a parabola in the plane xy, its vertex at the origin, its 
axis along the axis of at, and / its latus rectum; RP the generating 
parabola in any position with its plane parallel to zx, vertex in OR, 
and axis parallel to Ox, and let /' denote its latus rectum. Also let 
ON = X, NM =^v, MP =: be the coordinates of any point P in it; 
also draw RM' parallel to Oy. 



6o NOTES ON SOLID GEOMETRY, 

Then z' = l\ RM = l'. M'N and / = I'.OM' ; 
butOM^ + M'Nzn.r. 

.*. ~--\-—-^=ix (103) the equation to the surface. 

97. 7^ determine the form of the elliptic paraboloid from its equation. 
Since only positive values of x are admissible, no part of the sur- 
face is situated to the left of the planej'^. But the surface extends 
indefinitely in the positive direction of jv. If we makej^' = o, 2:*= I'x 
is the equation to the principal section OQ, and all sections parallel 
to zx are parabolas equal to OQ, with vertices in OR ; similarly, all 
sections parallel to xy are parabolas equal to the other principal sec- 
tion OR, with vertices in OQ. If we make x — h we have 

f z' _ 
Ih'^l'h-^' 

Therefore the sections parallel to ^are similar ellipses, and hence its 
name. 

Cor. If /=3 / the equation becomes x+z'^ = Zr, the paraboloid of 
revolution. 

98. To find the equation to the hyperbolic paraboloid. 

Definition. This surface is generated by the motion of a parabola 
whose vertex lies on a fixed parabola, the planes of the two parabolas 
being perpendicular to each other, their axes parallel, and their con- 
cavities turned in opposite directions. 

Let OR be a parabola in the plane of xy, vertex at the origin, and 
axis along with the axis of .r, and /its latus rectum, RP the generat- 
ing parabola in any position, vertex in OR, axis parallel to O.r, 
and let I denote its latus rectum, and ON = x, NM —y, MP = 0, 
the coordinates of any point P in it ; draw R*M' parallel to Oy, 
Then 

2^=/'. MR and/^/.OM'; 

but OM'-MRz= ON = .r. 
Hence : — — x (104), the equation of the surface. 

99. To determine the form of the hyperbolic paraboloid from its equation. 
The surface cuts the coordinate axes only at the origin, and since 

the equation admits positive and negative values of .v, >', z, as great 



NOTES 4DN SOLID GEOMETRY. 6l 

as we please, the surface extends indefinitely both ways from the 
origin. 

If we makej^^ = o we have z^ = tx the principal section, the para- 
bola OQ, with its concavity turned towards the left oiyz, and all sec- 
tions parallel to zx are parabolas .^qual to OQ with their vertices in 
OR. Making = we have^^ = Ix the parabola OR, and sections 
parallel to xy are parabolas equal to OR with vertices in OQ. 

If we make ^ = o we have the principal section in yz, 

z^/ 1 z=i ±y \/r or two straight lines through the origin; and for sec- 
tions parallel to yz making x = /i we have 

~ -T-r = I a hyperbola with its vertices in OR, and con- 

i/i In 

jugate axis parallel to Oz, For h negative the section becomes 

z^ v^ 

-777 — %- =1 a hyperbola with its vertices in OQ, and conjugate 
In In 

axis parallel to Ov. 

The surface has but one vertex, and consists of one sheet and one 
infinite axis. 



1 00. A symptotic planes io Ihe hyperbolic paraboloid. 
The equation — -7- = x may be written 



j;2 ^2 / rx\ 

—r~~j^ V ^ "^ — 2" ) which has for its limiting form 

when y and z become infinitely great with regard to x, -— = —7- , 

V Z V z 

or ^-= = ± — -, This represents two planes -^^ = H — t= and 
V/ a// V/ 'S/I' 

V z 

— — = -=. . through the origin and asymptotic to the surface. 

V / V / 

These planes contain the asymptotes to all the hyperbolic sections of 
the surface parallel to yz, 

loi . ITie elliptic and hyperbolic paraboloids are particular cases of the 
ellipsoid and hyperboloid of one sheet respectively when the centres of these 
surfaces are removed to infinite distance, 

x^ v^ z^ 
Take the equation -^ + 4r + -^ =1, and transfer the origin to 
a^ 0^ c^ 

6 



62 NOTES ON SOLID GEOMETRY. 

the left vertex of the axis la (— ^, o, o). (New coordinates being 
parallel to the primitive.) 

{x—af y \ ^^ — . -^^ y^ -L "^^ — ^'^ 

or multiplying through by a -^^ h -tf ± -t- =2.r (i), 

a a 

in which — , and — are the semi-latera recta of the principal sec- 
a a 

tions in xy and zx. Now make a = 00, and put — and — , 



a 



which remain finite, equal to / and / respectively. 
/. (i) becomes 

— ± -^ = 2jt:, the equations to the paraboloids. 

102. The equatmts of the surfaces of the second order which we have 
been studying are of the two forms 

A;tHBy + Cs^=D (i) 
^f^Qz'^kx (2) 

and we will show hereafter that all the surfaces of the second degree 
may by transformation of coordinates be included in these two forms. 

The first form (i) includes the sphere, ellipsoid, hyperboloids, 
cones of second order, elliptical and hyperbolic cylinders — which 
have centres. For if — .r, —y^ —z be written for {x, y, z) in (i) 
the equation is not altered, therefore for ever}^ point P {x, j', z) on 
the surface there is a point P' {—x, —y, —z) and PP' passes through 
the origin O and is bisected in O. 

Moreover, the coordinate planes bisect all the chords parallel to 
the axes perpendicular to these planes respectively and 2lx^ principal 
planes of the surface. 

The second form (2) includes the elliptic and hyperbolic parabo- 
loids and the parabolic cylinder which have a centre at an infinite 
distance. 

The planes j/0 and zx are principal planes of the two paraboloids, 
the other principal plane being at an infinite distance. 

Also both families may be represented by the equation 



NOTES ON SOLID GEOMETRY, 63 

the origin being at the vertex and A = o when the surfaces have 
no centre. 

Examples. 

1. Construct the sphere whose polar equation is 

r^asin^cos^. ^V^<f^^^^^ W^f ^^^^(f 

2. Find the locus of the point the sum of the squares of the dis-^^ ^^ ., /^ 
tances of which from n fixed points is constant/<^v<^/v /j=* 2^/^--'=^-'^'~^~- - '^ it 

3. Find the locus of the point the ratio of the distances of which ^ 

from two fixed points is constant. -^^^'^ ,>:v^v^y-^^ ^^-/Jv^ '^'1$:^l^^o 

4. Find the equation of the surface generated by the motion of a 
variable circle whose diameter is one of a system of parallel chords 
of a given circle to which the plane of the variable circle is perpen- 
dicular. 

5. The sphere can be represented by the simultaneous equations 

X =■ a cos q) cos d \ 
y =z a cos ^ sin ^ V • 
z ■= a s>ixi cp ) 

6. The ellipsoid may be represented by the equations 

x=^ a cos cp co^ 6 \ £%^^V|1 */ 
y = b cos ^ sin ^ >• • 
s = ^ sin (p ) 

7. The hyperboloid of one sheet may be represented by the equa- 
tions 

X -= a SQC o) cos 6 ) 4,y i, 
y z=z SQC cp Sin u > ' 
z = c tan q) ) 

8. The hyperboloid of two sheets may be represented by the equa- 
tions 

X =^ a sec q) ) 

y = d sin d tan q) V • 
z — c cos 6 tan q? ) 

9. A line moves so that three fixed points on it move on three 
fixed planes mutually, at right angles. Find the locus of any other 
point P on its line. . 



64 NOTES ON SOLID GEOMETRY. ? 

Solution : 

Let the three fixed planes be the coordinate planes {x\yyZ) the coordinates 
of P. A, B, C the points in which the line meets the coordinate planes oiyz^ 
xz, xy, respectively. ' Take VA=a, PB=^, PC=^, ON=;»r, NQ=jr, QP=2, 
<ACA'=r<p, <CBx=zB (CA' being the projection of CA on the plane xy and B' 
the projection of B on the axis of x). 

Then x=:a cos cp cos Q^yz=zd cos q) sin 0, z=:c sin (p, — and therefore the sur- 
face is an ellipsoid. 

10. Find the locus of a point distance of which from the plane xy 
is equal to its distance from the axis of (coordinates rectangular). 

11. Find the locus of the centres of plane sections of a sphere 
which all pass through a point on the surface. 

12. Find the equation of the elliptical paraboloid as a surface 
generated by the motion of a variable ellipse the extremities of whose 
axes lie on two parabolas having a common vertex and common 
axis and whose planes are at right angles to each other. 

13. Find the equation of the hyperbolic paraboloid as generated 
in a similar manner by the motion of a variable hyperbola. 

14. Construct the surface r sin ^ = a, ■ ' 

15. Find the equation to the surface d = ^7t in rectangular coor- 
dinates. 



CHAPTER IX. 
RIGHT LINE GENERATORS AND CIRCULAR SECTIONS. 

103. Surfaces of the second degree admit of another division, viz. 
into those which can he generated by the motion of a straight line 
and into those which cannot. This property which we have seen to 
belong to the cylinder and cone we shall now show to belong also to 
the hyperboloid of one sheet and the hyperbolic paraboloid. The 
ellipsoid being a closed finite surface does not possess this property ; 
nor the hyperboloid of two sheets, since that consists of two surfaces 
separated by an interval ; nor the elliptical paraboloid, since that is 
limited in one direction. 

104. Straight line generators of the hyperboloid of one sheet. 
The equation of the hyperboloid of one sheet 



^^ ^ b' 



r= I may be written 



X'' 



— I 



yL 
b' 



- (^r)e-r)-(-+i)(--i)=°- w 



Now (A) is satisfied by the pair of equations 
^x z 



- + - 
a c 



and also by the pair 



X z 
\a c 



X 

a 



6* 



-i 



w I + 



=— ^ 



y 



OT I + 



r 



(B) 



(C) 



66 NOTES ON SOLID GEOMETRY. 

And m being arbitrary equations (B) represent a system of straight 
lines, and all of these lie on the hyperboloid as the two equations 
together satisfy the equation to the hyperboloid. 

Similarly equations (C) represent another and distinct system of 
straight lines which also lie on the hyperboloid which is the locus of 
both systems, and we shall see the lines of either system may be used 
as generators of the surface. 

105. No two generators 0/ the same system intersect one another. 
For example take two of the system (B), 



"I — i=i-'i 



f + 7 ^---O-^)/ 

Combining the first equation of (i) with the first of (2) we obtain 

{m' — w") f I —-y] =0 01 y—b. 

Combining the second equation of (i) with the second of (2) we 
have 

{rn' ^ m") (^i + ^\ = o or y = ^ L 

These values for j^^ being incompatible the lines do not intersect. 

106. Any generator of the system (B) will intersect any generator of 
the system (C). 



Take 

m 



■(:-r)=--i 1 

^ \ (3) of system (B) 



NOTES ON SOLID GEOMETRY. 67 

(:--r) =«■'(- 1) 



m 

\a c I \ bj \ 

(4) of system (C). 



Eliminating x, j\ and z we obtain the identity m'm^' = m^ni!\ 
therefore the lines intersect. 

Hence, through any point of an hyperboloid of one sheet two 
straight lines can be drawn lying wholly on the surface, 

107. No straight line lies on an hyperboloid which does not belong to 
one of the systems of generating lines (B) d?r (C). 

For, if possiJDle, suppose a straight line H to lie entirely on the 
hyperboloid, it must meet an infinite number of generating lines of 
both systems (B) and (C). Let two of these (one of B and one of 
C) intersect H in two different points, we could then have a plane in- 
tersecting the surface in three straight lines, which is impossible since 
the equation is of the second degree. Hence no such line as H can 
lie on the surface. 

108. The hyperboloid of one sheet may be generated by the motion of a 
straight line resting on three fixed straight lines which do not intersect, and 
which are not parallel to the same plane. 

In the first place it is necessary that the motion of a right line 
which is to generate a surface should be regulated by three condi- 
tions. For, since its equations contain four constants, four condi- 
tions would fix its position absolutely ; with one condition less the 
position of the line is so far limited that it will always be on a certain 
locus whose equation can be found. 

Take then three fixed generating lines of the system (B), these do 
not intersect, nor are they parallel to the sam.e plane. Now, if a 
straight line move in such a manner as always to intersect these three 
straight lines, it will trace out the hyperboloid of which they are the 
generating lines. 

For the moving line meets the hyperboloid in three points (one 
on each of the fixed straight lines), and hence must necessarily lie 
wholly upon the surface. For the equation of intersection of a line 
and this surface being a quadratic equation, if satisfied by more than 
two roots, it is satisfied by an infinite number. The moving straight 
line, therefore, in its different positions, will generate the hyper- 
boloid. 



68 NOTES ON SOLID GEOMETRY. 

109. Lines through the origin parallel respectively to generators 0/ the 
systems (B) and (C) lie on the cone 

—Y + -TT — ~2" (i^y^ptotic to the hyperloloid. 
For this equation of the cone may be put in the form 



(f-7) (fn)= 



J/ y^ 
b' b' 



which gives two systems of lines through the origin lying on the 
cone, one system evidently parallel to the lines (B) and the other to 
the lines (C). 

no. The projection of a generating line of either system upon the 
principal planes^ is tangent to the traces of the surface on those planes. 

The equation of the trace of the surface on the plane zx is 

a" c' ■"^* 

The projection of the line of system (B) on xz 

c, ( X z\ X z m^-\-i x i — m^ z , . 

m^ [ I + -+ - =2m; or . - -\ • -=i (i). 

\acjac 2m a 2m c ^ ' 

X z 
Now, the condition that a line in the form - -f - = i shall be 

P ^ 

x^ z^ a^ CSL. 

tangent to the hyperbola -^ r —^ is -r- r=i. 

'' ^ a^ c^ p^ / 

This condition is fulfilled by the projection (i), for 



^m^ a^ ^m^ c'^ 



4 y m^ J 4 \ ^i J 2^2 



{m'-\-iy {i-m'Y 

Hence this projection is tangent to the hyperbola. 

III. The straight line generators of the hyperbolic paraboloid. 
The equation of the hyperbolic paraboloid 



NOTES ON SOLID GEOMETRY. 



69 



^ IT = *^ niay be written 



= X. 



And hence it is satisfied by the pair of equations 



V~i V/' 



= vix 



m 



or by the pair 






V Z 

^ H -z^ = mx 



(D) 



VI 



■s/ 1 V/ 



/ y_ !^' 



= 1 



(E). 



Hence the surface has two systems of straight line generators (D) 
and (E). 

The lines of both systems are parallel to the asymptotic planes 
of the surface respectively. The equations of these planes being 



y 



^ + 



V/ V/ 



= o and 



y 



V/ ^/l' 



== o. 



112. We can show in the same manner as in the Articles (34) 
and (35) that no two lines of the same system intersect ; and that a 
line of either system intersects all the lines of the other system, 'and 
that no other line than the lines of these two systems can lie on the 
hyperbolic paraboloid. And hence that through every point of the 
surface two lines may be drawn which lie wholly on the surface. 
And as in (108) that this paraboloid may be generated by the motion 
of a straight line which rests on two fixed straight lines and is con- 
standy parallel to a fixed plane ; also by a straight line which rests 
on three fixed straight lines which are all parallel to the same plane. 

113. The projections of the generating lines on the principal planes are 
tafigent to the principal sections of the paraboloid. 

The principal section in xy is^ = Ix (i). 



70 NOTES ON SOLID GEOMETRY, 

The projection of any line of the system (D) on xy is 

2y , I m^-J ' y7 
-—= — mx-\ or v = x-\ . (2) 

Now the tangent line to the parabola y=: Ix is of the form 

•v = /;tr -i : and if / = then — =: , 

-^ 4/ 24/ 2m 

Hence the projection (2) is tangent to the section y = Zr. 

114. Distinctions of sia' faces of secoiid order generated by straight 
lines. 

All the generators of the cone intersect in one point. All the gen- 
erators of the cylinder are parallel. Hence cones and cylinders are 
called 7'uled surfaces or developable surfaces. In the case of the hy- 
perboloid of one sheet and the hyperbolic paraboloid, the genera- 
tors of neither system intersect or are parallel. These are styled 
twisted or skew surfaces. The distinction between these last two sur- 
faces is that the generators in the paraboloid are parallel to a fixed 
plane. 

115. Plane sections of surfaces of the second order. 

If we intersect the surfaces represented by the general equation 

AjcH By + Cs- + 2A'.r0 + 2B>4- 2Ca:y + 2A'';v+ 2B'> + 2C''2 = D 

by the plane = we will obtain 

A^^ + B>'^ + 2C'jr)^-f2A"^ + 2B'> = D (i) a conic section. 

If we intersect it by a plane z =^ a we have for the curve of inter- 
section 

Kx' + Bf + 2Cxy-^2G'x-V2liy-\-'fpi.= T>\ 

a conic similar to the conic (i). 

Therefore sections of surfaces of the second order by parallel 
planes are similar curves, and hence, in determining the form of these 
sections we may confine ourselves to the discussion of sections through 
the origin. 



NOTES ON SOLID GEOMETRY. 



71 



116. To determine the nature of the curve formed by the intersection of 
a surface of the second order by any plane. 

Take the equation 

A.r^ + By^ + Cs^ = 2A'x And in order to get the equation of the 
curve of intersection in its own plane 

Make 

x^= X cos q) 4-y cos d sin cp 

y z= x' sin q? —y' cos 6 cos cp 
z =y sin 6, See Art. (71). 

Arranging the result we have 

x"^{K cos^<7? + B sin^ cp) + 2xy{k — B) cos 6 sin cp cos cp 

4-y^((A sin^ (p + B cos'^ cp) cos^ 6 -\-C sin^ 6) — 2AV cos cp 

+ 2h!y cos 6 sin cp, 

the equation to a conic section which will be an ellipse, parabola or 
hyperbola, (including particular cases of these curves,) according 
as the quantity 

(A-B)^ cos' e cos' cp sin' (??-(A cos' 9? + B sin^ ^)(A cos' 6 sin' cp 

+ Bcos' 6>cos' 9? + Csin' 6*) 

or — AB cos' cp—hC cos' <^ sin^ ^— BC sin^ cp sin' ^, (i) 

is negative, zero or positive. 

Hence every section of an ellipsoid is an ellipse because A, B and 
C are all positive. 

The sections of the hyperboloids may be ellipses, parabolas or 
hyperbolas since one or two of the quantities A, B and C will then 
be negative. 

For paraboloids A =r o. Hence for the elliptic paraboloid in 
which B and C have the same sizes the section is an ellipse ; except 
when ^ = o or (^ = o in which cases it is a parabola. 

For the hyperbolic paraboloid since B and C are of contrary signs 
the section is a hyperbola except when ^=0 or ^=0 when it is a 
parabola. 

117. Circular sections. Since the section is referred to rectangular 
axes it cannot be a circle unless the coefficient of x'y' vanishes 



>J2 NOTES ON SOLID GEOMETRY, 

or (A— B) cos 6 sin cp cos ^ = o 

n 7t 7t 

or C7 = — or 9:?= — , or ^ = o 

which shows th2it /or a circular sectmt the cutting plane must be perpen- 
dicular to 0716 0/ the principal planes of the surface, 

118. Let us now examine the surfaces -of the second order for cir- 
cular sections. 

Take first the surfaces having a centre and therefore represented 
by the equation 

A:v^ + B/4-C0^= I. (i) 
Since every circular section must be perpendicular to a princi- 
pal plane, let the cutting plane contain the axis of y^ and make 
the angle 6 with the plane xy— 

To transform (i) to this plane make 

X = x' cos d 

z — x' sin 6. Art {^2), 

Hence we have 

x'\\ cos^ (5 + C sin* 6) -f B/'^ = i (2) 

which represents a circle if 

A cos^ e+C sin^ (9 r= B 

_ ^ B-A , . 
or tan^ ^ = C^B' ^^^ 

We must now examine for each of the surfaces which axis it is 
that coincides with the axis ofj^. 

1°. For the ellipsoid A =— , B=-7^, C =— r 



Hence for a real 6 b must lie (in value) between a and c or the 
axis of the surface to which the cutting plane of circular sections is 
parallel is its mean axis. 

2°. For the hyperboloid of one sheet since we cannot have B ne- 



NOTES ON SOLID GEOMETRY. 73 

gative we must put A = — ^, B = ^ C = — ~ 

.-. tan l9 = ± -y -T-7-X2 
a r + 0' 

/. <5 > <3: or the cutting plane is parallel to the greater of the real 
axes. 

3°. For the hyperboloid of two sheets since we cannot have A and 
C negative, we must put 

A-— B-— C~- — 



,*. ^>r or the cutting plane is parallel to the greater of the im- 
aginary axes. 

Since tan has two equal values the cutting plane may be inclined 
at an angle 6 or i8o°— 6 to the plane of x}\ Hence there are two 
sets of parallel circular sections of the surfaces having a centre. If 
the surface becomes one of revolution we have tan ^ = 00 or o, and 
the two positions of the circular sections coincide with each other, 
and are parallel to the two equal axes. 

119. Secondly. For the surfaces not having a centre, we take 
equation By^ + Cz^ = 2Mx (i). 

1°. For the elliptic paraboloid, B and C have the same sign. 
Transforming (i) we have By^-^-Cx'^ sin^^= 2K'x cos 6] and hence 
for circular sections we must have the condition C sin^^ = B, or 

sin ^ == ± y — . Therefore the cutting plane is perpendicular to the 

principal section whose latus rectum is least. 

2°. For the hyperbolic paraboloid, since B and C have different 
signs, sin 6 is imaginary, and no plane can be drawn which shall in- 
tersect it in a circle. This was evident, too, from the fact (Art. 
116) that the hyperbolic paraboloid can have no elliptic sec- 
tions. 

120. Then, to sum up, all the surfeces discussed with the excep- 
tion of the hyperbolic paraboloid admit of two sets of planes of cir- 

7 



74 



NOTES ON SOLID GEOMETRY. 



cular sections. Therefore they can be generated by the motion of a 
variable circle whose centre is on a diameter of the surface. 

121. The planes of circular section may be found directly from 
the equations of the .surfaces, as follows: 

The equations of the central surfaces 

may be written B(:rH/ + :5') + (A--B)j^2-(B-C)0'=i; 

or 



B(^H/ + 2') + (VA-B ^+v'B^IC.2)(VX=B.;t:-y'B-C.0) = i 
which shows that either of the planes 



\/A-B;k:4- VB-C.0=:O (i) VA-B a:— VB--C.0 = o (2) 
^ . cuts the surface on which it cuts the sphere 

Hence the planes (i) and (2) and all planes parallel to them cut 
the surface in circles. 

The equation to the elliptic paraboloid may be treated in a similar 
manner, thus showing its planes of circular section. 

122. Sections 0/ Cones and Cylinders. 

1°, The sections of the cones may be inferred from Art. 95. For 
elliptic cones sections of the hyperboloids by any plane "i^ always ^'*^ 
similar to the section of the asymptotic cone to the surface made by 
the same plane, as is evident from the equations respectively. Hence 
the section of a cone of revolution by a plane will give an ellipse, 
parabola, or hyperbola. But we will examine this case more par- 
ticularly. 

In the equation of the cone of revolution 

r^ 
0(?'\-y^z=. — (z — cY, or x'^-\-y^ = tan^ v (^—0* 

(when - = tafil!) put x=x' cos ^ ' 

..,,,-,..: ^^=y ' 

-"■J ) J z = x' sin d ' 



NOTES ON SOLID GEOMETRY. 



75 



And we have for the curve of intersection by the plane containing 
the axis of^ 

y^(cos^ ^tan^ V — sin- 6) +y^ tan^ v-{-2cx' sin 6^c^ = o (i). 

This equation (i) represents an ellipse, parabola, or hyperbola, 
according as cos^ 6 tSjrl; — sin'* ^ is > = < o, that is according as 
tan ^<z=> tan v. 

2°. For the cylinder of revolution about the axis of ^, we make 
x=x' cos 6,y =y in its equation x^+y i= r^; ,\ the curve of in- 
tersection is x'^ cos^ 04-y^z=z r an ellipse. 

Examples (Coordinates Rectangular). 

1. Find the right line generators of the hyperboloid 

x^ f z' 

— + = 1 

9 4 1 ^^^ 

for the point (2, 3 ?) on the surface. - /"^ 

2. Find the right line generators of the paraboloid 4>^^-— 2 52:^=1 oo»r 
for the poinfc'(? 2, i) on the surface. j- 

3. Find the planes of circular sections of the following surfaces : 

6:^^ + 4rH9^'=36 (i) :^^S^'- o 

^x:'^^f-l^z'^\^^ (2) ^xtfx^r 
x'^zf^Az'=i2 (3) x^"^^:^':. J 
6x'fsy=3^^' (4) 

4. In the hyperboloid of revolution of one sheet — -~ 2" ~ ^ 

find the equations of the generating line whose projection on the 
plane xz is tangent to hyperbolic section in that plane at its vertex. 

5. Find the sections of the cone x^-\-y={z—2y by planes con- 
taining the axis of j^, at angles to the plane xy of 30°, 45°, and 60'' 
respectively. 

6. Find the curve of intersection of the surface 

x'+y+—=i 
4 

by a plane inclined at an angle of 30° to the plane xy, and whose 
trace on that plane makes an angle of 45° with the axis O^. 



CHAPTER X. 

TANGENT PLANES, DIAMETRAL PLANES, AND 
CONJUGATE DIAMETERS, 

123. Straight line meeiing surfaces 0/ second order. 
We can transform the general equation 

kx* + By' + C2''+2A>+2B'2a:+2C'.r>' + 2A";tr + 2B'> +2C"0 + 

F=o (I) 

to polar coordinates by writing x ^^ Ir, y ^=z mr, z = nr, (when /, m, n 
are in rectangular coordinates, direction cosines, and in oblique co- 
ordinates, direction ratios). The equation becomes 

/^(A/' + Bw^ + Cfi' + 2Ps.'mn + 2^' In + 2Ci??i) 

+ 2r(A'7+B";;z + C^^;/) + F= o. (2) 

Hence a straight line meets the surface in two points, and if these 
two points be coincident the line is tangent to the surface, 

124. Tangent Plane to surfaces of second order. 

Let the origin be on the surface (and therefore F=: o) then one 
of the values of r in (2) is r = o. Now, in order that the radius 
vector shall touch the surface at the origin, the second root must be 
o, and the condition for this is A'7+B'V;z + C;/ = o. Multiplying 
this by r and replacing /r, mr^ nr by x,yj z, this becomes 

A";, + B> + C''^ = o. (3) 

Hence the radius vector touching the surface at the origin lies in the 
fixed plane (3); and as /, m, n are arbitrary, A";t: + B"_;; + C"0 = o is 
the locus of all the radii vectores which touch the surface at the 
origin, and is therefore the tangent plane at the origin. 

Hence, if the equation of the surface can be written in the form 
«2 4-Wi= o (where u^ represents terms of second degree and u^^ terms 

76 



NOTES ON SOLID GEOMETRY. 



77 



of first degree in x, y, and z), then u^=z o is the equation of the tan- 
gent plane at the origin. 

Therefore, lo find the equation to the tangent plane to the surface at the 
point x'y'z, transfer the origin to this point. The equation may then he 
written Ug 4- Uj = o^ a7id Ui == o is the tangent plane referred to the 
point of contact as origin ; then in Ui= o retran.fer the oj'igin to the 
primitive one, 

125. For the central surfaces (origin at centre) take the equation 

and let {x\ y\ z') be the point of contact. Transferring the origin to 
the point {x', y\ z') by the formulae 

X = x + x \ 

y =y + y > we have 

z=z z + z') 

Ax' + B/ + Cz' + 2kxx' + 2^yy + iCzz' = o. 

Hence the tangent plane at the new origin is 

kxx' -f ^yy + Qzz'=. o. (i ) 

Now retransfer the origin for equation (i) to the centre by the 
formulae 

x ^=^ x — x^ \ 

y = y —y' )• and we obtain 

z = z — z' ) 

Axx'-^-Byy + Czz- Ax"-- By''- Cz"= o, 

or Axx' + Bjy + Czz=i (2) the required equation of the tangent 
plane, at the point xy'z' referred to centre. 

I 



1°. For the sphere A r=B r= C 



a 



Hence (2) gives xx +yy + zz'= a^. (3) 

2°. For the ellipsoid A = — ,Bi=:-7^C= — 

a^ b^ c^ 

xx^ yy' zz' , , 

3"". For the hyperboloid of two sheets A =: — B:=: 0= -^ 

a^ 0^ r 



XX yy zz , . 



7* 



78 NOTES ON SOLID GEOMETRY. 

4^ For the hyperboloid of one sheet, A==— r-, B=-rx-. C= — -s. 

126. For the surfaces which have no centre (origin at vertex) by 
treating the equation By^ + C-s:^ = 2h!x in a similar manner we 
obtain 

Byy + Czz^ = A\x-\-x') (7) for the equation to the tangent plane 
to the elliptical paraboloid and 

B)y—-Czz=:A'{x-\-x') (8) for the' tangent plane to the hyper- 
bolic paraboloid. 

Remark. The same method may be applied to cones and cy- 
linders. 

127. Polar planes io surfaces of second order. The equations (3) 
(4) (5) (6) (7) (8) are the equations to the polar planes to the sur- 
faces respectively with respect to the point {x\ y\ z) and these polar 
planes possess properties analogous to the polar lines to the conic 
sections. 

128. The length of the perpendicular from the centre on the tangent 
plane io the ellipsoid is p = \/a^ cos^ a + <^'^ cos'^ /^ + c^ cos'^ y , when 
cos ay cos ^, cos y are its direction cosines, 

Ihe equation to the tangent plane is — r + '^ + — ^ = i. It may 

also be written x cos a -\-y cos /3 -\- z cos y =p* Hence we must 
have 

/ __ cos a _ cos /3 __ cos y __ci cos a _b cos ft ^c cos y __ 

i~ x' ~~ y "" z' ^ x' ~~ y ~ z' "~" 



— A /^^cos^a + <^^cos^/^ -I- f^cos^;/ ___ ^ 

— /i/ ^^^2 -7^ ^7^ ' — V^^cos^a 4- b^cos^fi 4- Aos'y . 

Hence calling the direction cosines /, m^ n^ the equation of the 
tangent plane may be written 



lx^my-\-nz=i ^ aH' -f- b'^m^ + c'n^ . ( 9 ) 



NOTES ON SOLID GEOMETRY. 



79 



129. To find the condition ihai the plane 



h -4- H =1 (i) shall be tangent to 

a /3 y ^ ^ ^ 



x^ / z^ 
the ellipsoid » -y- + "tt "^ r = ^* 

^ . / V . , -'t'Jt:' j^' zz^ 

Comparing (i) with — f + "ir + ~F = ^ 

we must have 

'^~"^' ^'^~¥' y"""^ 
°' ^-^="^' 7"="!' y=~' •'• «q"^""^^^d 

a^ 3^ c^ 
adding ~T + ~^ "J 2" ~ ^ is the required condi- 

tion. 

130. The sum of the squares of the perpendiculars p, p', "^"^ from the 
centre of the ellipsoid on three tangent planes mutually at right singles is 
constant. 

Let cos a cos yS cos y\ cos a! cos /J^ cos y\ etc., be the direction 
cosines. 
Then /^ _ ^2 ^^^gS ^ ^32 ^^gu /J +^2 ^.^gS ^ 

/'^ = c^ cos^ a:' +<^^ cos^ /5' +^^ cos^ /' 
/^2 = a^ cos' ^^' + 3' cos' y3" + ^*^ cos' /', 
and adding we have 

131. Cor. Hence the locus of the point of intersection of three tangent 
planes to the ellipsoid which intersect at right angles is a concentric sphere 
of the radius ^ d^ + 3' + c'. 

For «' the square of its distance from the centre is equal to 
/+/'+/'', and therefore to a^ + 3' + ^'. 

Remark. In the case of hyperboloids one at least of the quantities 
a', ^', c^ is negative, and hence their sum may be negative or nothing ; 
in the former case there is no point in space through which three 
rectangular planes touching the hyperboloid can be drawn, and in 
the latter case the centre is the only point which has that property. 



8o NOTES ON SOLID GEOMETRY. 

132. Diametral Planes, Definition.' A diametral surface is the 
locus of the middle points of a series of parallel chords of a given 
surface. Diametral lines or diameters are the intersections of the 
diametral surfaces. 

133. To find the diametral surface corresponding to ce^ given series of 
parallel chords in a surface of the second order which has a centre. 

Let the equation of the surface be 

Ajt^ + B/ + a^==i, (i) 

7,772, n the direction cosines of each the parallel chords, and x\ y\ z' 
the coordinates of its middle point. 
The equation of the chord will be 

x—x __ y—y' _ z—z _ 
I m n ' 

Then for the points in which it, meets the surface (i) we shall 
have 

K{x'^lry-^B{y'^mry-^C{z'-Vnry=i', 

or {KP + Bw' + Cn^y + 2{klx' + Bmf ^- Cnz')r + Kx"" + By" + Cz"'^ i. 

Imposing on this the condition of equal roots for r, we have 
Klx' -{-Bmy' -\-Cnz' ^=^ o (2) the equation of the diametral surface, a 
plane passing through the centre. 

X V z 

1^4. The diameter — = ^ = — is one of the series of parallel 
I m ft 

chords bisected by the plane (2), and is called the diameter conju- 
gate to the plane, and conversely the plane Ix + my -^^ 7iz == o is con- 

^ ^. A.r Bv Cz 

jugate to the diameter -r-= -^ = — . 
I m n 

If a diametral plane be chosen as a new plane of xy and its con- 
jugate diameter be taken as the new axis of 0, the centre O being still 
the origin ; then, since every chord parallel to Oz is bisected by 
the plane xy, the equation of surface will contain only the second 
power of 0. Hence, if there be three planes through the centre the 
intersection of any two of which is conjugate to the third, the equa- 
tion of the surface referred to these planes will be of the form 

AV+By + CV=i, (3) 

that is of the same form as the equation referred to rectangular axes. 



NOTES ON SOLID GEOMETRY. 3 1 

135. To find the condiiions that of three planes through the centre 0/ a 
surface of the second order each may he diametral to the intersection of the 
other two. 

Let the planes be 

lx-{-my-^nz^=^ o, Tx-\-my-\'n\ = o, I'^x + m'y + n'z = o. 
The equations of the diameters conjugate to the first plane are 

Ax _ By _ Csr 
I m n ' 

and if this be parallel to the other two planes, we shall have 

,, I , m , n , ,// / ,f ^^ „ ^ 

these with the third equation l^ —r- + w' -5-+ ^'"tt- = o, found in 

like manner, are the required conditions. 

These three planes are called conjugate planes, and their intersec- 
tions conjugate diameters. 

Since we have only three relations between the six quantities there 
will be an infinite number of systems of conjugate planes in each 
surface. 

136. Equations referred to conjugate diameters. If in (3) Art. 134 we 
make 

A'—— "R^-- JL C' — — 

Then for the ellipsoid 

x^ y^ z" 

-7-2+773 + — 2= I will be the equation referred to conjugate di- 
ameters, and a! , b\ c will be the semi-conjugate diameters. 
For the hyperboloids we shall have 

-^-^--^- I and ^^^^l-^ I 

Remark. The tangent planes at the extremities {x\ y', z) of any 
diameter to a central surface are parallel to the diametral plane 
conjugate to the diameter so that the conjugate plane of the diameter 
through the point {x\ y, z') on the ellipsoid is 

xx' \y' zz' 



82 NOTES ON SOLID GEOMETRY. 

137. The sum 0/ the squares 0/ three conjugate semt'diameters oy the 
ellipsoid is co?tstant. 

In the first place^ any point on the ellipsoid may be represented by 
the equations x ^=^ a cos A, j^ = 3 cos ^, z ^=^ c cos v, when cos A, 
cos yu, cos V are the direction cosines of some line, for the condition 
cos^ A + cos^ yu + cos^ y = i cause these three equations to satisfy the 
equation of the ellipsoid. 

Therefore if cos A, cos /i, cos r, cos A', cos //', c6s r' are the direc- 
tion cosines of two lines answering to the extremities of two conju- 
gate diameters, these will be at right angles to each other. 

ror the equation — ~ +^^ — r^ o will give 

cos A cos A' + cos )jL cos /i' + cos y cos y' = o. 

Now the square of the length of any semi-diameter .v'^4-y^+ 2'''^ 
expressed in terms of A, //, y, is 

a"^= a^ cos- \^V cos^ }x-\-c^ cos* y, 
and of the conjugates in terms of A', }x\ y\ \'\ fx", y'* 
l"^—a^ cos^ A' + ^- cos^ }Ji^ ■\-c^ cos^ y' 
c"=: a' cos^ \"-{-P cos' }x"-\-c' cos' y". 
Adding we have 

a"" ^h''' -\-c"^ = a'^ + P -\-c\ since the lines A, //, y, \\ //', i^^ and 
A", ^'\ y" are mutually at right angles. 

138. To find the locus of the intersection of three tangent planes at the 
extremities of three conjugate diameters. 

The equations of the three tangent planes are 

X ^ V z 

— cos A + v cos u-{- - cos r = I 
a b c 

— cos A + ^ cos y -f - cos y = i 
a c 

X V z 

-COS A''+ ^cos y" H — cos y" = i. 
a c 



Squaring and adding, we get for the equation of the locus 

x' 

— H 
a^ 



—^ + ^4-— 7-= 3 an ellipsoid with the semi-axes a^/~i ^ !^V~Tf 



^v-' 



NOTES ON SOLID GEOMETRY. 83 

139. The parallelopiped whose edges are three conjugate semi-diameters 
of an ellipsoid has a constant volume. 

Let Ox, Oy, Oz be the semi-axes of the surface a, b, c\ Ox\ Oy , 
0-s' any system of semi-conjugate diameters a\ b\ c; let the plane 
of x'y' intersect that of xy in the semi-diameter 0^i= A, and let 
0^2= B be the semi-diameter of the curve Xi x' which is conjugate to 
0^1. Hence parallelogram a'b'=i parallelogram AB. 

.-. Vol {a\ b\ c')=Yo\ (A, B, /) 

for these figures have the same altitudes and equal bases. 

Let the plane z'Oy^ intersect xy in the semi-diameter Oy^ = C, 
then this plane must contain Oz ; for, being conjugate to O^t'i in a 
principal plane it must be perpendicular to that plane ; hence Ox^, 
Oj'i, Oz form a system of semi-conjugate diameters, and any two of 
them are semi-conjugate diameters of the plane section in which they 
are situated. k QJ 

.-. Vol (A, B, c') =Vol (A, C, ^ a*C 

Vol (A, C, ^) =Vol {a, b, c) ^ 

.'. Vol {a\ b\ c') =:Vol {a, b, c). 

140. To fi?td the diametral plane bisecting a given system of parallel 
chords in the case of the surfaces which have not a centre^ 
Taking the equation of the surface 

B/4-a^=2A":v 

A r^u I, J ^r— jc' V—y' z — z' 

and one of the chords — - — = ^ =— - = = r 

I m n 

the equation of the diametral plane will be 

m By-i-n Cz = l^ 

Hence the diametral planes are parallel to the common axis of the 
principal parabolic sections. 

We cannot, therefore, in these surfaces have a system of three con- 
jugate planes at a finite distance, but we can find an infinite number 
such that for two of them each bisects the chords parallel to the other 
and to a third plane, by proceeding as in Art. (135). 

By taking the origin where the intersection of these two meets the 
paraboloid, and referring to these three planes, the equation of the 
surface will be of the form 

By + Cz'=2E''x. 



84 NOTES ON SOLID GEOMETRY. 

And the third plane is evidently the tangent plane to the surface at 
the new origin. 

141. The tangent planes io the hyperlohid of one sheet and the hyperbolic 
paraboloid at a point x'y'z' intersect the surfaces each in two right line 
generators through the point of contact. 

The equation of the hyperboloid of one sheet referred to any con- 
jugate diameters is 

a" "^ b" c" ~^'* 

and the equation of the section made by any plane y ^=- ft parallel 
to the conjugate plane oi xz, is 

a:' c" ""^ b"'' 

and it is evident that the value /?= y gives us the section of the 
tangent plane at the extremity {x\ y, z^) of the diameter y-, or 

x'^ z^ 

— 7^ 7^ = o, two right line generators. 

For the hyperbolic paraboloid 

By_CV=2E'';t' (i) 
the tangent plane through the origin is ^ = o, and its intersection 
with (i) is 

B^— CV= o, two right line generators. 



CHAPTER XL 

GENERAL EQUATION OF THE SECOND DEGREE IN 

x,y, AND z, 

142. In order to discover all the curves represented by the general 
numerical equation 

-^— Ax^ + 2AV+2A"^ 

W +2B'2x + 2B'y 
Cz' +2Cxy-}-2C'z = D (E) 

we will first transform the coordinates to a new origin by means of 
the formulae 

X =. a + x^ ] 

y = ^+y [ (I), 

z ^ y-i- z' ) 

and endeavor to determine the coordinates (<t, /3, y) of the new 
origin in such manner as to cause the terms of the first degree to dis- 
appear. If this can be effected the equation will be reduced to the 
form 

kx" ^W -\-Cz^+2k'zy -\- 2Wzx + 2Cxy =¥' (F) 

in which there is no change when -x, —y, — are substituted for 
+ .V, +j', -tZ, and which therefore represents a surface having a 
centre, and the new origin of coordinates is at this centre. 

Now, several different cases may arise according to the numerical 
relations among the coefficients A, B, C, A', B', C, A", B", C". 

1°. a, /?, y the coordinates of the centre may each have a finite 
value found from the three equations determining the conditions of 
the transformation. 

2°. a, ft, y may have infinite values. 

3°. a, ft, y may be indeterminate. 

8 85 



86 NOTES OX SOLID GEOMETRY. 

The surfaces corresponding to these three cases will be 

(A) Surfaces having a centre. > 

(B) Surface's having no centre (centre at an infinite distance). 

(C) Surfaces having an indefinite number of centres. 

143. Making the actual transformation of (E) by the formulae (r) 
we have 

A^H2A;rs + 2(A^ 4_C^/5-fB';/+A")jc 
By + 2B'0.v + 2(C'a' + B/S +A';k + B^^)v 
Qz" +2avr+2(B'a' + A'/3+Cx+C'> 



' ByS'^ + 2B'a';/+2B"/? - = D. 
-^2Ca^ + 2C'y) 



[Cy 



And in order that the terms of the first degree in .v, y, z shall dis- 
appear, we must have 

A« + C'/54-B>-f A"-= o \ 
C'a + B/? + Ay + B"=o ^ (C) 

B'«f + A'/J + C;/ + C"=o ^ 

which are called the equations of the centre. 

1°. If these three equations give finite values for a, fj, y, then 
the surface represented by the given equation has a centre. 

2°. If two of these equations are incompatible this shows infinite 
values for a, /3, y, and the surface has no centre. 

3°. If the three equations reduce to two, then the surface has a 
line of centres. For each one of the equations is the equation of a 
plane, and two taken simultaneously represent a line, and the surface 
is an elliptical or hyperbolic cylinder. For, cut the surface by the 
planes P and Q, P cutdng the line of centres (D) and Q containing 
that line. The section by P is a curve of the second degree having 
its centre on the line D, ard hence an ellipse or hyperbola. The 
section Q will be two straight lines parallel to the line D, and as Q 
may revolve about D in all its positions giving two straight line sec- 
tions parallel to D, the surface is a cylinder. 

4°. If the three equations reduce to a single one, then the surface 
has a plane of centres (/. e., the given equation represents coincident 
or parallel planes). 

Note. The equations of the centre can be found in any given 




NOTES ON SOLID GEOMETRY. 87 

equation most readily by finding the derived equations with regard 
to x^ y, and z respectively {t, e., by differentiating with regard to x, j', 2 
respectively), the x, y, and z in the resulting equ'ations standing for 
a, /J, y. 

144. Example i. Determine the class of the surface represented 
by the equation x'^ + 3^ + 4s^ + 2yz + ^zx + 6xy^26x—2^y—'^2z=26, 

The equations of the centre are 

2x-{-^z-^6y — 26 = o 
6y -\'2z -\- 6x — 24 := o 

8s + 2;F + 4jv — 32 — O 

and the surface has a centre. 

Example 2. Determine the class of the surface 

x'^ -{-y^ — 2z'^ + 2yz + 2XZ + 2xy^4x— 2y -f 20— o. 
The equations of the centre are 

2X-{-2Z-{-2y~-4. = o ) 

2y-\-2Z-\-2X—2 — O ^ 

— \Z-\-2y-\-2X-\-2 = O ) 

the first two of which x-\-yi-z = 2, x-\-y-\-z =: i are incompatible, 
hence the coordinates of the centre are infinite, and the surface has 
no centre. 

Example 3. Determine the class of the surface 

x'^ + 4y —z^ — 2yz —%x -f \xy +20 = 0. 
The equations of the centre are 

20:— 2: +4>' = o \ 
Sy—2z-^4x = o V. 
— 2 z — 2y—x-{- 2=0) 

The first two of these are identical, hence the three equations re- 
duce to two and the surface has a line of centres (/. e., is a cylinder). 

Example 4. Determine the class of the surface 

Sx^ + 1 8>'^ + 2z^ -hi2yz-{- Szx + 24;r>'~ 50.^—757—250 + 75=0. 

The equations of the centre are 

i6:v-h 80 + 24_y— 50 = o ) 
36 y+ 120+ 24.V — 75 = o ,- 

404- I2l'+ S.V— 25 =: O ) 



88 



NOTES ON SOLID GEOMETRY. 



which are all three the same, each being 8.v+ I2r-f 42^ = 25. Hence 
the surface has a plane of centres, and consists of a pair of parallel 
planes. 

145. Recurring to the general equations of the centre 

C«'+B/J + Ay + B"= o '> (C) 
B'a' + A'^ + C;K + C"= o \ 

we may find an easy rule for a relation among the coefficients in any 
given equation by which we can distinguish the central surfaces from 
those having no centre and those having an infinity of centres. 

The common denominator of the values of a, f5, and y in these 
equations is the determinant 



R = 



A, 


C, B' 


c, 


B', A' 


B', 


A, C 



: AA"^ + BB - + CC'"^ - ABC - 2 A'B'C 



Now, if R be different from zero, the surface has a centre ; but if 
R = o it may either have no centre or an infinity of centres. 

The value of R mav be written out bv the followins: mnemonic 



form 



A, 


B, 


c; 


A' 


B' 


c 


A' 


B' 


c 



the letters to be multiplied by columns for the first 
three terms, and by rows /or the two last. 



146. To find an easy rule for F, the nezv absolute term m the trans- 
formed equatio?i of the central surfaces when the origin is moved to the 
centre. 

This complete transformed equation is 

A.V' + By 4- CV -f 2^zy -f 2^'zx + 2C xy — F when 

I Art'^ + 2A'/^;K + 2A"«f \ 
F=D- ^^ B/5'^-i-2B^a;K + 2B"^ '^ . 

Now, multiplying the first of the equations (C) of the centre by 
a, the second by ft, and the third by y, and adding them 



we have 



\ B/5'+2B'ay + B"/3 - = o. 

{ Cy' + 2C' a ^ + C"y ) 



NOTES ON SOLID GEOMETRY, 89 

Hence F=D-(A"a + B''/5-hCV). Therefore the rule for F is 
substitute for x, y, z in the tei'vis of the first degree one-half thd co- 
ordinates of the centre (i, e., ^a, |-/J, ^y respectively) and take result 
from D. 

Example i. Taking the Example i, Art. (i44),in which the coordi- 
nates of ihe centre are found to be jc =1, j^ = 2, 0=3, 
we have F=26 + 26 x ^-f-24 x i +32 xf =1 1 1; 

and the transformed equation is 

x^ + 3y^ + 4<s^ + 2yz 4- ^zx -f- 6xy == 1 1 1 . 
Ex. 2. 2x'^ + ^y'^-\-^z^-{-^yz-\-6xz + ^xy—6x—'Sy—i^z— 20. 
Here the coordinates of the centre are x = i,y — 2, z='—i. 
,\ F=20 + 6 x^4-8 X I + 14 X — I=::I7; 
and the transformed equation is 
C" 2x^ + 3/ + 42^ + 8yz + 6xz + 4^y = 1 7. 

147. Removal of the terms in xy, xz, yz. Reduction of the equation of 
ihe second degree to two forms. 

For a more complete discrimination of the surfaces represented by 
the general equation, we will now remove the terms in xy, xz, yz by 
a transformation of coordinates. So far we have made no supposition 
as to the direction of the axes. Henceforth, for convenience, we will 
consider the axes rectangular. 

' Taking the equation (E) in rectangular axes we propose now to 
transform it to a system also rectangular in such manner that the 
terms in xy, xz, yz shall disappear. The disappearance of these 
terms can only be effected by taking for coordinate planes either dia- 
metral planes or planes parallel to them. 

. We will therefore begin by finding a diametral plane conjugate to 
a given diameter. 

148. To find a diametral plane conjugate to a given diameter, 

x—a y—b _ z—c 
I m n 

Putting x = a-\-lr, y = b-^mr, z =: c + nr in the general equation, 
and arranging with reference to r, we have for the coefficient of the 
first degree in r 

2{Al+B'm + Cn)x+2{Cl-i-Bm-{-A'n)y-\-2{B'l+A'm + Cn)z 

■j-2{A''l+B''m + a'?i) =0; 



90 



NOTES ON SOLID GEOMETRY, 



and this placed equal to zero is the equation of the diametral plane, 
namely 

(A/+B';;z + CV/).v + (CV+B;7z+A'4j'+(B7+A';;/ + 0/)2 + A'7 

4-B";;z + C";/ = o. 

149. To determine a diametral plane perpendicular to the chords which 
it bisects, that is, to find a principal plane. 

In order that the diametral plane shall be perpendicular to the line 

.V a V h Z—C . . J. . r ^r^^ 1 

— - — — = , we must have the conditions fulnlled 

/ m n 

A/+C';;/ + B';^ _ Cl-vV>m-\-Mn _ B7+AV;/+C; ^ 
/ in 71 

or putting each of these equal to s. 

A/+B';/+C'/?/ = /r \ 
C7+B7;^ +A';/ = ms V (A) 
B7-f AV;/ + C;/ = ns ) 

and also the condition P -\- 7?r -\- n"^ — i , 

To determine /, w, and ;/ in equations (A) we first find s. Writ- 
ing these equations 

(A--j^)/+C';;/ + B';/ = \ 
Cl-\-{B-s)m + A'n = o I 
B7 + A';;/ + (C-^);/ = 

they give the result 

A~^, C, B' 
C, B-.r, A' 
B', A', C-s 

or (A-S)(B-S')(C-S)-(A-S)A'^ + A'B'C -C'^(C-S)-hA^B'C 

_B'^(B-S;=:o; 

or 

S'3_(A + B + C)/ + {AB + AC + BC-A'^-B'^-C'^)^4-AA'^ + BB"^ 

4_CC^'^-ABC-2A'B'C'=o (D). 

This cubic has necessarily one real value for s, which substituted 
in (A) gives one set of real values for /, m, n. Hence there is one 
principal plane. 

For convenience of discussion let us take this plane perpendicular 



NOTES ON SOLID GEOMETRY, 



91 



to the axis oi z, then / r= o, m = o, and nz=L\, And hence equa- 
tions (A) give ^' — o, A'=: o, and the general equation transformed 
to this principal plane as plane o{ xy is of the form 

Now we know from the like discussions in conic sections that one 
transformation is always possible, and but one to a system of rectan- 
gular axes in the plane xy which shall cause the term in xy to dis- 
appear. Hence there are three principal planes, and three sets of 
values for /, m^ n, and the cubic (D) has three real roots. 

The general equation may then be always reduced in rectangular 
coordinates to the form 

L.r24-My + N22 + 2L^^ + 2M> + N0=D. (E^ 
which represents then all the surfaces of the second order. 

150. JVie reduction 0/ this equation Lx^ + My^ -f-Nz^ + 2L'x-f 2M y 

-f 2N z r= D to two forms. 

1°. If L, M, and N are different from o. 

Then we may cause the terms of the first degree to disappear by 

. . ^ . , u . . L' , M' N' 

transiernng the ongm to the pomt x ^^— -=— , J' = ^^ ^ ^ =■ — irr- 

The surface will then have (jc', y , z) for its centre, and the equation 
will be of the form 

Kv2_^Mj;2^N0^Z= F. (I.) 

2^. If one of the three coefficients, L, M, N, for example L = o 
and L^'^be diff"erent from o. 

We cannot then cause the term 2L7^ to disappear, but by trans- 
ferring the origin to the point 

D , M' , N' . Ml 1 u r 

X — - — , r = — ^rr, -2^ = ^i:^ the equation will take the form 

2L - INI N ^ 

M/ + NV=2V.v. (II.) 

The forms I. and II., we have seen, belong to the surfaces of the 
second order, which we have already discussed. Hence the general 
equation of the second degree (E) represents these surfaces and no 
others. 



92 



NOTES ON SOLID GEOMETRY, 



151. The form I. we have seen represents the ellipsoid, the two 
hyperboloids and cones of second degree, and includes the elliptic 
and hyperbolic cylinder, My^ + N^^ =F and parallel planes NV=F. 

The form II. represents the elliptic and hyperbolic paraboloids, 
and the parabolic cylinder. 

152. The complete reduction 0/ the equation of the seco7id degree to the 
simple forms \, and II. Use of the discriminating cubic (D). 

The resolution of the equations (A) furnishes each value of j* in 
the cubic (D), one system of values of/, m, n. We have then three 
systems, /, m, n\ /', in\ n\ /", ;;/', n\ which are the direction cosines 
of the three rectangular axes {^principal axes) to which the surface 
must be referred in order to cause the products jit, a-^, >'-2: to dis- 
appear ; the formulae of transformation are then 

X = Ix -Viy , -{-Tz' 
y =1 VI x + m^y ■\-m"z 
z = nx^ -\-ny -\-n'^z\ 

If we take only the terms in x'^ in this substitution we find 
L = A/^ + B;;^HC«' + 2A'ww-f 2BW+2C7/;/. 

But if we multiply the equations (A) respectively by /, m, n and add, 
remembering that l^ -\- 7?i^ -[- n^ z=z i we have 

A/HB;;z2-f CV + 2A'w« + 2BW+2C7w = s; 

Hence L is a root of the cubic (D) and M and N are the other two 
roots. 

For the values of L', M', N' we will have 

V =A"/ +B''m -\-C'n \ 
I\r= A'7' ■\-^"m'^C'n' \ (M). 
N' = A^r + B'V/' + C'V ) 

The absolute term D does not change in this transformation since 
the origin is not changed thereby. 

For the surfaces having a single centre after solving the cubic, we 
liave only to calculate F, for which we have given a rule. 

For the surfaces having no centre the coefficient designated by V 
is equal to — U, and is computed by first finding in equations (A) 
the values of /, w, n^ which correspond to S = o. Both in the cases 
of surfaces having no centre and a line of centres, one root of cubic 
= o and we have only a quadratic to solve to determine L and M. 



NOTES ON SOLID GEOMETRY. 



93 



153. For surfaces having a centre, if we wish only to discover the 
particular class of the surface, without making the complete trans- 
formation of the equation to its centre and axis, the sign of the roots 
of the discriminating cubic will tell us whether the surface is an ellip- 
soid, hyperboloid of one sheet, or hyperboloid of two sheets. These 
signs we can ascertain from inspection by Descartes's rule * without 
solving the equation. 

Example. Find the nature of the surface "jx^ ■\-()}'^ ■\-^z^—\yz— \xy 
1= 6. The cubic (D) gives 

S' — (7 + 6 + 5)S'^+(43 + 35 + 30— 4— 4)S-|-28-f-20— 210 = o; or 
S^— 18S- + 99S— 162 =: o. 

.'. The row of signs is H 1 , three changes of sign. Hence 

all the roots are + and the surface is an ellipsoid. 

So also for surfaces having a line of centres, the signs of the roots 
of the quadratic into which the discriminating cubic degenerates, 
serve to distinguish the elliptic from the hyperbolic cylinder. 

And for surfaces having no centre, the signs of the roots distinguish 
the elliptic paraboloid from the hyperbolic paraboloid. 

154. Recapitulation of the viethod of reduction of iiumei'ical equations 
of tJie second degree and of distinguishifig the surfaces represented by them. 

We now^ propose to give the mode of distinguishing the nature of 
the surface represented by any given numerical equation of the second 
degree in x, y, and 0, and of finding its principal elements. 

I. Form the equations of the centre, and also the discriminating- 
cubic from the remembered form 
S3_(A + B-hC)S^ + (AB + AC + BC-A'^-B'^-C'^)S + AA'^-fBB'^ 

4_CC'_ABC-2A'B'd=: o, 

observing that the absolute term is equal to R, the denominator 
of the values of the coordinates of the centre in the general equation, 

ABC 
and therefore can be formed by the mnemonic A'B'C (Art. 145). 

A'B'C 

Then 

155. 1°. If R be different from o, the surface has a centre. Find 



Note. ''All the roots being real the number of positive roots is equal to the number of 
changes of sign in the row of signs of the terms, and the number of negative roots is equal to the 
number of continuations of sign." 



94 



NOTES ON SOLID GEOMETRY, 



the coordinates of the centre and transform to the centre by the rule 
in Art. (146). Determine the signs of the roots of the cubic by Des- 
cartes's rule. Then calling these roots L, M, and N, and calling F 
the new absolute term on the second side of the equation. 

Then 

a. If L, M, N all have the same sign as F, the surface is an ellip- 
soid. 

h. If L, M, N all have a different sign from F, the surface is im- 
aginary. 

f. If two only of the roots L, M, N have the same sign as F, the 
surface is the hyperboloid of one sheet. 

d. If only one of the roots L, INI, N has the same sign as F, the 
surface is the hyperboloid of two sheets. 

e. If F = o and L, ]\I, N all have the same sign, the locus is a 
point. 

f. If F = o and one of the roots L, M, N has a different sign 
from the other two, the surface is an el|ip^ cone (Art. ^^), 

156. 2°. If R = o the cubic has one of its roots S = o and is 
degraded to a quadratic, the coefficient of S, namely AB + AC + BC 
—A'^ — B'^ — C"^, becomes the absolute term. 

And if the equations of the centre are incompatible the surface has 
no centre. 

Then 

a. If the roots M and N of the quadratic (degenerate cubic) have 
the same sign (/. e,) if AB + AC + BC- A'''-B''-C''> o the surface 
is the elliptical paraboloid. 

b. If M and N have different signs (/. e.) if AB + AC + BC-A'"^ 

— B'^ — C'"<o the surface is the hyperbolic paraboloid. 

c. If one of the roots M or N be zero {i. e.) if AB + AC-f BC-A'' 

— B'^— C"^= o the surface is the parabolic cylinder. 

^157. 3°. If R rr o and the equations of the centre can be reduced 
to two equations, the surface has a line of centres. The cubic as in 
(2°) has one of its roots S = o and degenerates into the quadratic 

S--(A + B-f-C)S + AB + AC-hBC-A'--B'«-C"'=:o. 

Then 

a. If the roots M and N of this quadratic have the same sign {i. ^.) 



NOTES Oy SOLID GEOMETRY. 



95 



if AB + AC + BC — A"- — B'- — C'"^ > o the surface is an elliptic cy- 
linder. 

b. If the roots M and N have different signs {i. e.) if AB + AC 
4-BC — A'^ — B'* — C'^< o the surface -.is the hyperbolic cylinder. 

c. If in the reduced equation of the cylinder M2:^ + N)^^ := H, H be 

equal to o, and M and N both of same sign, the locus is a straight 

z — o\- 
hne \ . 

<^. If H = o-and ^\ and N be of different signs the surface con- 
sists of intersecting planes. 

158. 4°. If R == o and the equations of the centre become a 
single equation, the surface has a plane of centres, and consists of 
two parallel or coincident planes, which are readily found by solving 
the equation with reference to any one of the variables. 

159. 5°. In the case of surfaces of revolution the cubic has equal 
roots. To examine the cubic for equal roots in the case of central 
surfaces of revolution, we simply look for 9, common root between it 
and its first derived equation (differential). 

160. General Remark. In any of the above cases we may com- 
plete the reduction by solving the cubic to get the new axes and 
thus obtain their direction by finding /, m, n from equations (A), 
And in the case of the surfaces without a centre we may find V, 
from equations (M). 

161. Remark I. In the cases of surfaces having a line of centres 
and of those not having a centre, we can distinguish readily the sur- 
face represented b) a given numerical equation through sections by 
the coordinate planes. 

1°. If the equations of the centre show a line of centres, sections 
by tne coordinate planes will tell whether the surface is an elliptic or 
a hyperbolic cylinder. 

2°. When the equations of the centre show no centre, then 

a. If there are ellipses among these sections by the coordinate 
planes, the surface is an elliptical paraboloid. 

I. If there are hyperbolas among these sections, the surface is a 
hyperbolic paraboloid. 

c. If all these sections are parabolas, or one of them parallel 
straight lines, the surface is a parabolic cylinder. 



96 



A'Ol^ES OiV SOLID GEOMETRY, 



162. Remark II. Again, if the terms of the second degree in the 
given equation break up into unequal real factors, the surface must 
be either the hyperbolic paraboloid or hyperbolic cylinder, and these 
two surfaces are otherwise readily distinguished. We may note also 
that if the terms of the second degree in the given equation form a 
perfect square, the surface is either a parabolic cylinder or two 
parallel planes. 

163. We will now illustrate by a few examples : 

Ex. I. 7.r^— I3>^' + 62-+24A3'H- 1210— I22.r rr: ±84. 

As this is a central surface with the origin at the centre, we only 
need the discriminating cubic,, which is 

/ — 343^^ + 2058 = o ; or /±or — 343^^^+ 2058 — o. 

The signs + ± h show one coniimiatmi and two changes, and 

hence the surface is a hyperboloid of one sheet, or two sheets, accord- 
ing to the sign of 84. 

By trial we find that 7 is a root of the cubic, and then by depress- 
ing the equation we find the other two roots are 14 and —21. There- 
fore the equation of the surface referred to its centre and axes is 

y.v^-!- i4v^ — 2 10'=: ±84; or .r^-f 2^^—30-=: ± 12, 

Ex. 2. 2^X'^+2 2y-{-l6z'^-\-l6)'Z — 4ZX—20X)'—2 6x — 40V — 443 

= —44. 
The equations of the centre are 

25.V— iqi'— 20 =: 13 

— io.r + 22^+ 80=20 

— 2A'+ 8j'+ 160 = 2 2 ; 

whence we find the coordinates of the centre x = i,y =1, 0=1. 
Moreover 

F=- 44 + 26 . J + 40.i + 44-i= 9- 
The discriminating cubic is 

j^ — 65/+ 1134^ — 5832 = o. 

Its signs give three changes. Hence all the roots are positive. 
The surface then is an ellipsoid. 



NOTES OjV solid GEOMETRY. 



97 



By trial we find that 9 is one of the roots of the cubic. Hence the 
other two are 18 and 36. The reduced equation is then 

()x^ + 1 8y -f 362^ = 9 ; or .r^ + 2y^ + 42^ == i . 
And the principal semi-axes are i -— , _ . 

V2 2 

Ex. 3. 5.r^+ ioy+ I'jz^ -\- 26yz+iizx + i^xy -\- 6x -\-iy -\- loz =64. 
The equations of the centre are 

^x+ T^H- 90 =—3 
7.r+ioy+ 13s =—4 
9.v+i3_>'+i70 =-5. 

Multiplying the first of these equations by — i and the second 
by 2, and adding, we obtain the third. Hence the equations are 
only two independent ones. The surface is therefore a cylinder. In- 
tersecting it by the coordinate plane xy, i, e., making 0=0, we obtain 

z^x^j^iAxy+ioy''-\-6x-\-^y= 64, 

which is an ellipse. The surface is therefore an elliptic cylinder. 
To complete the reduction we transfer the origin to the point 

r /Y> _1_ rj \f j QCr — 'J '\ 

where the line of centres u y o { pierces the plane 

9:r+i3j^/-f 170= — 5 ) "^ 

X, y, that is, to the point = 0, ^ = 1, x =:—2^ 
and find F=:64+6— 4 = 66. 

Also the discriminating cubic is 

j^^ — 32/4-6^- = o, which gives /— 32^^ + 6 = o, 

the roots of which are 16 + 5 V 10 and 16 — 5 V' 10. 
And the reduced equation of the cylinder is 

Ex. 4 . 5^^ + 5/ + 82^ + 40>' + ^zx — ^xy + 6x + 6y—7,z — o. 
The equations of the centre are 

5.V-47H-20 =—3 \ 

x-i- y-{-4z = I ) 



V 
^8 NOTES ON SOLID GEOMETRY, 

Adding the two last of these equations we have 
Sx—4y-\-2z =— 2^. An equation w^hich is incompatible with the 
first. Hence the surface has no centre. 

The cubic /- i8/ + 8ij'= o; or s''—i8s-}-8i = o, 

which gives two roots equal to 9. The surface is therefore a para- 
boloid of revolution 

gy^ + gz^ = 2 Yx. 

To find V, we first determine /, m and n. For these we have the 
equations 

4/—^m—2n = o 

/+w + 4« = o 

22 I 

which give / = — , w = — , n =z . 

3 3 3 

Therefore (Eq. M) A'"= 3 . - -f 3 .-+ - . ~ = - 

33232 

and 2V =— 2 A'" =—9. 

The reduced equation of the surface is therefore 

Ex. 5. 2Ayz-i-2B'zx + 2Cxy + 2K"x-\-2B"y+2C"z — D. 

The cubic is 

S3-(A'2 4.B'-^ + C'^)S-2A'B'C = o. 

The surface is a hyperboloid if A', B' and C are all different from 
o. If A'B'C is of the same sign as F in the reduced equation the 
cubic will have two roots of the same sign as F and the surface will 
be a hyperboloid of one sheet. In the opposite case it would be a hyper- 
holoid of two sheets. 

If A' = o the cubic becomes 

S^ — (B''^ + C'^) =0, whose roots are of different signs. Hence 
the surface 2B' zx ■\- 2C xy ^- ih!' x -^ 2B"y ^ 2C z = o is a hyperbolic 
paraboloid. 

Ex. 6. x^ +y 4- 92^ + 6>'2 — ()xz — 2xy + 2x—az ■= o. 

The equations of the centre are incompatible and the terms of the 



NOTES ON SOLID GEOMETRY. 



99 



second degree form a perfect square, hence the surface is a parabolic 
cylinder. 

Examples. 

164. I. Find the nature of the surfaces represented by the follow- 
ing equations. 

(i). I ix^ -\- sy -^ ^^' — 20yz -{- 4ZX + i6xy + 22X ■{- i6y i- 4z -{- 11=0. 

(2). X' ^y^ + z^ + 2JZ + 2XZ -h 2xy — lox— loy — loz -h 2 ^ = o. 
^(3)- 3x'^—sy'^—i2yz-\-i2zx-\'8xy—6x—6y-{-^z = c. ^o./z-^a *- 

(4). 4x^ + qy^-^gjz'^—-i6zx-{'^^zy='36, ^ '-/^^v z":^* 3 r% £t^^Z. 

(5). 3JC-+ 2^—2X0 f 4^2 — 4^^ — 80— 8 =: O. 

2. The equation 7;t'^4-8y + 4-s^— Z>'<2r— ii^jv— 7^j/ = ^^ represents 
a hyperboloid of one sheet. 

3. The equation A:^+y^+3-s:V 3j'-24-2:j;+^— 7^—147—250= 13— ^ 
represents an ellipsoid, a point, or an imaginary surface according 
as ^is < = > 67. 

4. The equation x'^+y^-^-z^ +yz+zx + xy = a^ represents an oblate 
spheroid. 

5. Find the nature of the surface {y'-zy+{z—xy + \:f—jA- =si^-. 

6. Find the nature of the surface >'0 + zx-i-xy = d^. 

7. ax^-\-\y^-^^z^^i2yz^()zx-V\xy-{-\\x-\-i6y^2\z-{-\'] r= o re- ^ 
presents an elliptic paraboloid, a parabolic cylinder or a hyperbolic 
paraboloid according as a > = < i, 



CHAPTER XIL 

PROBLEMS OF LOCI. 

165. Prob. I. To find the surface of revolution generated by a right 
line turning around a fixed axis which it does not intersect. 

Let the fixed line be the axis of z and let the shortest distance a 
from the revolving line to the axis of ^ lie along the axis of x in the 
original position of this line so that its equation is .r — a, j/ = mz. 

Then the equation of the surface is 

X -{-y = a' -^7?rz' 

or — 2 ¥~= I- 

a^ a^ 

The hyperboloid of revolution of one sheet. 

Prob. 2. To find the locus of a poi?il whose shortest distances from two 
given non-intersecting, non-parallel straight lines are equal 

Take the axis of z along the shortest distance between the two 
lines, the plane xy perpendicular to z at the middle point of this 
distance 2c, and the axes oi x and y bisecting the angles between the 
projections of the line on their plane. Then the equation of the lines 
will be 

z ^^ c \ z — —c 
y = mx ) ' y z^ — mx 

and we have (z — cY^^^ ~^— (z-^cY^-— t— 

^ ' i^ni^ . ^ I + VI 

or 

cz{\ ^nf) ^mxy ^:^ o, a hyperbolic paraboloid since it has no 
centre and its term of second degree breaks up into two real factors. 

Prob. 3. Two planes mutually perpendicular , contain each a fixed 
straight line. To find the surface generated by their line of intersection. 
Take the axes as in Prob. 2. Then the equations of the planes are 

100 



NOTES ON SOLID GEOMETRY, 10 1 

Y^{z—c)'^-y—mx — o\ (i) Y^{z-\-c)-^y-^mx — o. (2) 
The condition of perpendicularity of these planes is 

KK'-j-i—w^ = o, and eliminating K-f-K' between this equation 
and equations (i) and (2) we have 

y~ wV + ( I '-ni^)z^ = ( I - rn^Y 
which represents a hyperboloid of one sheet. 

Prob. 4, To find the surface generated by a right line which always 
meets three fixed right lines no two of which are in the same plane. 

For greatest simplicity take the origin at the centre of a parallelo- 
piped, and let its faces be at the distances a, b, c respectively from 
the coordinate planes ^0, xz, and xy. Then take three edges of this 
parallelopipedon as the three fixed lines fulfilling the conditions. 

yz=b ] z = c \ X = a ] . ^ 

Assume for the equations of the movable line 

X —x' _ y —y^ _z — z^ . 

cos a cos ^ cos y' ^ 

The conditions that the line (4) shall meet the lines (1) (2) and 
(3) are respectively 

y^ — b z'-\-c z — c x' -\- a x — a y' -h b 

cos fi cos y ' cos y cos a ' cos a cos fi ' 

Eliminate the arbitraries a, /?, y by multiplying the equations to- 
gether, and we have for the surface 

(^x—a){y^b){z-c)z={x-¥a){y + b){z^-c)', 
or reducing 

ayz + bzx + cxy + abc == o, . 

which the discriminating cubic shows to be a hyperboloid of one 
•sheet. The same surface will be generated by a straight line resting 

, , , , jr = a ) v=: — b ) X =^ — a ) 

on the other three edges > , } . , } , 

z = — c ) z = c ) y ^^ \ 

Prob. 5. To find the surface generated by a right line which always 



I02 NOTES ON SOLID GEOMETRY, 

meets three fixed right lines ^ no two of which are in the same plane, but all 
of which are parallel to the same plane. 

Take one of the fixed lines as the axis of x, and then the other two 
parallel to the plane oi xy. Then their equations are 

Now, the equations of a moving line meeting lines (i) and (2) are 
* _ , / z\ r (4) (^ ^^d ^ arbitrary), and the condition that this 
line shall also meet (3) is Ic = mk (c—d), 
and eliminating the /and k by means of equation (4), we have 

cy mx{c—b) 
~V'^ z-b ' 

or cyz ■\-m(b— c)xz — cby = o, 

a hyperbolic paraboloid, as its equation shows no centre, and the 
terms of the second degree break up into two real factors. 

Prob. 6. To find the sujface generated by a right line which meets 
V^'O fixed right lines, and is always parallel to a fixed plane. 

Since the two fixed lines must meet the fixed plane, we can take 

c (0, "~ : (2), as in 2, as the fixed lines, and the 

2:3=^ \ ^ ' z^=. — c ) 

plane V2 as the fixed plane. 

Then the equation of the moving line parallel \.o yz 

is Z / ( (3)> ^' A ^^d k arbitrary. 

The conditions that this line shall meet the lines (i) and (2) 

mk =z Ic+p 
are 

— ?;//(' — : — lc-\-p ] 

or mk = Ic 2indp =^ o ; 

or eliminating /, k, and /, 

y 

mx z= c ~ ; 
z 

or mxz = cy, a hyperbolic paraboloid. 



NOTES ON SOLID GEOMETRY, IO3 

Prob. 7. 2 wo finite non - intersecting non -parallel right lines are 
divided each into the same number 0/ equal parts ; to find the surface 
which is the locus of the lines joining corresponding points of divi- 
sion. 

Let the line which joins two corresponding extremities of the given 
lines be the axis of 2: ; let the axes oi x and^ be taken parallel to the 
given lines and the plane oi xy be halfway between them. Let the 
lengths of the given lines be a and b. 

Then the coordinates of two corresponding points are 

2: zz: r, X ^ ma, y z= O] s = —c, x ^=^ o, y ^= mb ] 
and the equations of the lines joining these points are 



x y ") 

ma mo 

r ' 

J 



2X Z ^ 

ma c 



whence eliminating m the equation of the locus is 

2cx^a{z^c) (f+^) 
a hyperbolic paraboloid. 

Prob. 8. To find the locus of the middle points of chords of a surface 
of the second order that has a centre, which all pass through a given fixed 
poiftt. 

Take the given point for the origin and two conjugate diametral 
planes which pass through it for the planes of zx and xy, and a plane 
parallel to the third conjugate plane for that o^yz; then the equation 
to the surface will be of the form 

ax'' + by^ 4- cz'^ + 2a 'x +/ = o. 

Let X ^1 VIZ, y = nz he the equations of any chord. Combining 
these with the equation of the surface, we have 

(am^ + bn^ -\-c)z^+ 2a'' viz + ^ = o, 

in which the values of ^ belonsr to the extremities of the chord. 



I04 VOTES ON SOLID GEOMETRY, 

Therefore the z of its middle point is 

,_ a"rn 
^-' am'-^bn' + c ^^^' 

and the other two coordinates of the middle point are 

x'=z mz\ (2) y^nz\ (3) 

Hence eliminating m and n the required equation of the locus 

a surface of the second order similar to the first, and passing through 
its centre and through the origin. 



CHAPTER XIII. 
SOME CURVES OF DOUBLE CURVATURE. 

1 66. To find the equations to the equable spherical spiral. 
Definition, If a meridian of a sphere revolve uniformly about its 

diameter PP' while a point M moves uniformly along the meridian 
from P to P', so as to describe an arc equal to the angle through 
which the meridian has revolved, the locus of M is the equable 
spherical spiral. 

Taking PP' as the axis of 0, PAP' the initial position of tke plane 
of the meridian as the plane oi xz^ the equation of the sphere is 

x'-Vf^z'=.a\ 

Let POM — 6, AON = q), then, by definition 6 =q), 
and from polar coordinates 

X = a cos 6 cos q), y ^=: a cos ^ sin q)] 
,\ X = a cos^ 6, y = a cos 6 sin 6. 

Therefore ^'^+y = a'^ cos^ 6 (cos'^ ^4-sin'^ 6) =: ax. 

Hence the equations of the spiral are 

x'+f + z' =a' {i) x'-Vf = ax ; (2) 

or the spiral is the curve of intersection of the sphere and a right 
circular cylinder whose diameter is the radius of the sphere. 
If we subtract (2) from (i) we obtain 

z^ = a^ —ax (3) a parabolic cylinder. 

And the equations (2) and (3) also represent the curve, which is 
therefore also the intersection of a right circular and right parabolic 
cylinder at right angles to each other. 

167. To find the equations to a spherical ellipse. 

Definition. The spherical ellipse is a curve traced on the surface 

105 



/ 



Io6 NOTES ON SOLID GEOMETRY, 

of a sphere such that the sum of the distances of any point on it 
from two fixed points on the sphere is constant. 

Let SH be the two fixed points on the surface of the sphere 
whose radius is r, C the middle point of the arc of the great circle 
which joins them. If P be any point of the spherical ellipse SP and 
HP arcs of great circles, then 

SP + HP = 2a = a constant. 

Through P draw PM, an arc of a great circle perpendicular to 
SH, and let SH = 2;/, CM = cp, PM = B. 

Then, in the right-angled spherical triangle SPM we have 

cos SP = cos {y — (p) cos d. 

And in the triangle HPM 

cos HP = cos (y—cp) cos 6. 

C-D. XTT. /SP+HP\ /SP-HP\ 
Now, cos SP -f cos HP = 2 cos f j cos f j 

/SP-HF 

= 2 cos a cos 

\ 2 

A A uTD QP • /SP + HP\ . /SP-HP\ 
And cos HP— cos SP = 2 sin ( ) sin ( j 

SP-HPN 

2 sin a sin ' 



Therefore, 

/SP— HP\ __ cos y cos cp cos 



^SP-H_P). 



COS 



\ 2 



Sin a 



^SP— HPN^ _^ sin y sin q) coe 



\ 2 / sin a: 

Squaring and adding 



COS'' y ^ sin^ V . o « /i 

— r-^ cos' m cos' u -\ — r—, — sin^ m cos^ c^ =1; 
cos' a ^ sin' a 

or if we transform from polar to rectangular coordinates 

cos^ y „ sin^ J^ . / x / / 
cos' a sin' ^^ \] 

This equation and the equation of the sphere'N 

^'+y-f^' = r2 (2) 

determine the spherical ellipse, as the intersection of a right elliptic 
cylinder and the sphere. 



NOTES ON SOLID GEOMETRY. 



107 



168. To find the equations to the helix. 

Definition. Whilst the rectangle ABCM revolves uniformly about 
its side AB, so that the parallel side CM generates the surface of a 
right circular cylinder, the point P moves uniformly along CM, and 
generates a curve called a helix. 

Let AB be the axis of 0, and when the rectangle is in the plane 
xz let P and M both be at D on the axis of x, and let the velocity of 
P = « times the velocity of M. 

.-. PM = «.arc DM. 

Also let AN = X, NM ^y^ PM = be the coordinates of P, and 
AM = a the radius of the circular base of the cylinder in the plane xy, 

X 

. •. z =. na cos~^ — , and j/^ + o;^ = a^ ( i ) 

are the required equations of the helix. 

Or we may represent the curve by the two equations 

X V 

z ^:z na cos~^ — , z =^ na sin~^ — (2); 

or the same in the forms 

z z 

X ^^ a cos — . y =1 a sin — , (3) and 

na ' -^ na ^^^ 

z / z \ z / z 

since cos — = cos [2m7T-\ ] and sin — = sin [2m7t-\ 

na \ na J na \ na 

the same values of x and y correspond to an infinite number of 
values of z. The equations (i) (2) and (3) show that the projec- 
tions of the helix on the planes xz^ and^^ give the curve of sines, 
and the projection on xy is the circle. 



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